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LIGHT 



PRINCIPLES AND EXPERIMENTS 



BY 

GEORGE S. MONK 

Assistant Professor of I J hy#'ic* 
University of Chicago 



FIRST EDITION 



McGRAW-HILL BOOK COMPANY, INC. 

YORK AND LONDON 
1937 



COPYRIGHT, 1937, BY THE 
McGiiAW-HiLL BOOK COMPANY, IN<\ 



PRINTED IN THE UNITED STATES OF AMKRICA 

All rights reserved. This book, or 

parts thereof, may not be reproduced 

in any form ivithout permission of 

the publishers. 



THE MAPLE PRESS COMPANY, YORK, PA. 



PREFACE 

l^ETring thirteen years' teaching of the subject of light at an 
intermediate level, in classroom and laboratory, the author 
has had the usual experience of finding it necessary to refer 
students to several different textbooks for outside reading to 
supplement the lectures. Rarely has it been possible to find a 
single textbook in which the treatment of a given topic contained 
the degree of elaboration consistent with the purposes of an 
intermediate course. For this reason the author believed that 
a text covering the essentials of geometrical and physical optics, 
with the addition of several chapters covering the more recently 
developed subjects of modern optics, would serve a useful 
purpose. 

The book is intended for students who have finished the equiva- 
lent of an ordinary sophomore college course in general physics. 
It is .intended for both those for whom an intermediate course 
in the subject is the last, and those who expect to continue 
graduate study in the field of light or in associated fields in 
the physical or biological sciences. For this reason, while the 
emphasises on physical optics, particularly interference, diffrac- 
tion, and polarization, considerable space has also been devoted 
to geometrical optics, a subject which is only too often not a 
familiar one to students who will later use optical instruments 
whose principles they should understand. A working knowledge 
of elementary mathematics, including the fundamentals of the 
differential and integral calculus, is required,- but so far as 
possible each topic has been treated so that abstract mathematical 
development is subordinated to the discussion of the physical 
concepts involved. This has required that in several instances 
where the mathematical theory is beyond the scope of the book 
only the results are set down, while in other cases mere algebraic 
development has been relegated to appendices. An experiment, 
not necessarily novel, has been tried in basing several of the 
problems upon illustrations in the book, thus supplying a. degree 
of substitution for laboratory experience. 



vi PREFACE 

Other texts have been drawn upon freely in compiling this 
one, principally Drude, "Theory of Optics"; Wood, "Physical 
Optics"; Preston, "Theory of Light"; L. W. Taylor, "College 
Manual of Optics"; Mann, "Manual of Optics "; Born, "Optik"; 
Williams, "Applications of Interferometry"; Hardy and Perrin, 
" Principles of Optics"; and to a lesser extent many others. The 
author acknowledges with gratitude advice and criticism by 
his colleagues, especially Professors H. G. Gale, A. H. Compton, 
and Carl Eckart, each of whom read parts of the manuscript. 
Thanks are also due Dr. Rudolf Kingslake for valuable criticisms 
of an earlier draft of the chapters on geometrical optics, and 
Dr. J. S. Campbell for criticisms of an earlier draft of the chapters 
on interference, diffraction, and polarization. Helpful criticism 
by Dr. George E. Ziegler, Mr. Richard W. Hamming, and Mr. 
Alfred Kelcy is acknowledged, as well as comments and correc- 
tions by members of classes during the preparation of the 
manuscript. A great deal is due to the helpful and stimulating 
advice given by Professor F. K. Richtmyer, who suggested 
important changes and additions. Acknowledgments for illustra- 
tions copied or otherwise obtained froni others are for the most 
part made at the point of insertion. Exceptions are: Fig. 7-8, 
which was copied from a cut kindly supplied by the Bausch and 
Lomb Optical Company; Fig. 11-17, which is a copy of a photo- 
graph made for the author some years ago by Dr. J. S. Campbell; 
Fig. 13-9, from a wash drawing made by Miss Libuse Lukas; 
Fig. 14-10a, from a spectrogram made by Mr. Leonard N. 
Liebermann; Fig. 16-1, supplied by the Mount Wilson Observa- 
tory; and Fig. 16-12, adapted from an illustration by F. E. Foster 
in the Physical Review, 23, 669, 1924. 

Finally, no words of the author can express the thanks due 
his wife, Ardis T. Monk, for criticisms of the manuscript, for 
reading and correcting the proof, and for the preparation of the 
index. ft 

GEORGE S. MONK. 

UNIVERSITY OF CHICAGO, 
September, 1937. 



CONTENTS 

PAOK 

PREFACE v 

CHAPTER I 
FUNDAMENTAL CONCEPTS IN GEOMETRICAL OPTICS 1 

Fundamental Postulates The Ray The Optical Length of a Ray 
Fermat's Principle The Principle of Reversibility The Law 
of Malus-VThe Focal Length of a Thin Lens-^l^o J"hin Lenses 
The Concept of Principal Planes Equivalent Focal Lengths. 

CHAPTER II 
THE LAWS OF IMAGE FORMATION 8 

Ideal Optical Systems Refraction at a Spherical Surface 
The Collinear Relation Lateral Magnification Collinear Equa- 
tions for a Single Refracting Surface Principal Points and Planes 
Conjugate Rays and Conjugate Points LaGrange f s Law 
Longitudinal Magnification Angular Magnification, Nodal Points 
Mirror Systems. 

CHAPTER III 

V, 
COMBINATIONS OF OPTICAL SYSTEMS 19 

Equation for a Thin Lens Combination of Two Systems A 
General Lens Formula Classification of Optical Systems 
Telescopic Systems. 

CHAPTER IV 

APERTURES IN OPTICAL SYSTEMS 31 

The Stop The Aperture Stop Entrance and Exit Pupils The 
Chief Ray Telecentric Systems. 

CHAPTER V 
PHOTOMETRY THE MEASUREMENT OF LIGHT 36 

Photonwftric Standards Brightness of Extended Sources Lam- 
bert's Cosine Law Photometric Principles Applied to Optical 
Systems Numerical Aperture Natural Brightness Normal 
Magnification Effects of Background. 

CHAPTER VI 

ABERRATIONS IN OPTICAL SYSTEMS 45 

Spherical Aberration Third-order Corrections to Spherical 
Aberration Coddington's Shape and Position Factors Astig- 

vii 



viii CONTENTS 

PAGK 

matlsm Primary and Secondary Foci Astigmatic Difference 
Coma Elimination of Coma Aplanatic Points Curvature of 
Field Distortion Chromatic Aberration Cauchy's Dispersion 
Formula The Fraunhofer Lines Two Kinds of Chromatism 
Achromatizing of a Thin Lens Achromatism of the Huygens 
Ocular The Secondary Spectrum. 

CHAPTER VII 
OPTICAL INSTRUMENTS. 72 

The Simple Microscope The Magnifier Compound Magnifiers 
The Gauss Eyepiece The Micrometer Eyepiece The Com- 
pound Microscope Numerical Aperture Condensers Vertical 
and Dark Field Illuminators Telescopes The Reflecting Tele- 
scope Oculars (Eyepieces) The Huygens Eyepiece The Rams- 
den Eyepiece Erecting the Image The Spectrometer. 

CHAPTER VIII 
THE PRISM AND PRISM INSTRUMENTS , 88 

The Prism Spectrometer Dispersion of a Prism-^Resolving Power 
of a Prism The Constant-deviation Prism Index of Refraction 
by Means of Total Reflection The Abbe Refractometer. 

CHAPTER IX 
THE NATURE OF LIGHT 100 

Light as a Wave Motion Velocity, Frequency, and Wave-length 
Simple Harmonic Motion Phase and Phase Angle Composi- 
tion of Simple Harmonic Motions Characteristics of a Wave 
Motion The Principle of Superposition The Wave Front The 
Huygens Principle; Secondary Waves Amplitude and Intensity 
The Velocity of Light Wave Velocity and Group Velpcity. 

CHAPTER X 
INTERFERENCE OF LIGHT 120 

Interference and Diffraction Compared Conditions for Inter- 
ference No Destruction of Energy Methods for Producing 
Interference Younfe's Experiment The Fresnel Mirrors The 
Fresnel Biprism The Rayleigh Refractometer The Williams 
Refractometer. 

CHAPTER XI 
INTERFERENCE OF LIGHT DIVISION OF AMPLITUDE 1 37 

^ Colors in Thip Fi|ma N^wtflTi'ff B^gffl Double and Multiple 
Beams Tbp> Minfrelftmi T^.piforometer The Form of the Fringes 
The Visibility of the Fringes, Visibility Curves Multiple 
Beams The Fabry-Perot Interferometer Intensity Distribution 
in Fabry-Perot Fringes Resolving Power of the Fabry-Perot 
Interferometer The Shape of the Fabry-Perot Fringes. 



CONTENTS ix 

PAGE 
CHAPTER XII 

DIFFRACTION. . . ^ 1G4 

Fresnel and^raunhofer Diffraction Fresnel Zonies The Zone 
Plate Cylindrical Wave Front DiffractioTT by a Circulai 
Obstacle Diffraction at a Straight Edge The Cornu Spiral 
Fresnel and Fraiifrfrofer Diffraction Compared Fraunhofei 
Diffraction by a Single Slit By Two , Equal Slits Limit oi 
Resolution The Stellar Interferometer Many Slits. TheLPif 
fraction Gratinp The Dispersion of a Grating Resolving Powei 
of a Grating The Echelon Diffraction by a Rectangular Open- 
ing Diffraction by a Circular Opening. 

i^ "' ' ' " ' ' "^^ ** 

CHAPTER XIII 
/POLARIZATION OF LIGHT 208 

Polarization by Double Refraction The Wave-velocity Surface 
Positive and Negative Crystals. Uriiaxial Crystals Polariza- 
tion by Reflection Brewster's Law Direction of Vibration in 
Crystals Plane of Polarization The Cosine-square Law of 
Mains The Nicol Prism Double Image Prisms. The Wollaston 
Prism Elliptically Polarized Light Wave Plates The Babinet 
Compensator The Reflection of Polarized Light Rotation of 
the Plane of Vibration on Reflection The Nature of Uripolarized 
Light The Fresnel Rhomb General Treatment of Double 
Refraction Optic Axes in Crystals Axes of Single Ray Velocity 
Rotatory Polarization FresnePs Theory of Rotatory Polariza- 
tion The Cornu Double Prism Half-shade Plates and Prisms. 
^ 

CHAPTER XIV 

SPECTRA 250 

Kinds of Spectra Early Work on Spectra The Balmer Formula 
for Hydrogen The Rydberg Number Series in Spectra The 
Hydrogen Series The Quantum Theory of Spectra Kirchhoff's 
Law of Emission and Absorption Kirchhoff's Radiation Law 
Stefan-Boltzman Law Wien's Displacement Laws Distribution 
Laws Planck's Quantum Hypothesis The Rutherford Atom 
Model The Bohr Theory of Spectra Energy-level Diagrams 
Band Spectra of Molecules Continuous Absorption and 
Emission by Atoms The Structure of Spectral Lines The 
Broadening of Lines The Complex Structure of Lines. 

CHAPTER XV 

LIGHT AND MATERIAL MEDIA 272 

Absorption Laws of Absorption Surface Color of Substances 
Color Transmission Absorbing Blacks Early Theories* of Dis- 
persion The Electromagnetic Theory of Dispersion The Quan- 
tum Theory of Dispersion Residual Rays Metallic Reflection 
The Optical Constants of Metals The Scattering of Light by 



x CONTENTS 

PAGE 

Gases Polarization of Scattered Light Fluorescence Polariza- 
tion of Fluorescence Phosphorescence Fluorescence in Gases 
Resonance Radiation Raman Effect The Photoelectric Effect. 

CHAPTER XVI 

THE EFFECTS OF MAGNETIC AND ELECTRIC FIELDS 300 

The Zeeman Effect Classical Theory of the Zeeman Effect 
The Anomalous Zeeman Effect Quantum Theory of the Anoma- 
lous Zeeman Effect The Stark Effect The Faraday Effect The 
Kerr Magneto-optical Effect The Kerr Electro-optical Effect 
The Cotton- Mou ton Effect Measurement of Time Intervals 
with Kerr Cells Velocity of Light with Kerr Cells. 

CHAPTER XVII 
THE EYE AND COLOR VISION 323 

The Optical System of the Eye Defects in the Optics of the 
Eye Binocular Vision The Stereoscope Optical Illusions 
The Contrast Sensitivity of the Eye Flicker Sensitivity, Per- 
sistence of Vision Spectral Sensitivity Color Hue Saturation 
Brilliance Color and the Retina Complementary Colors 
Theories of Color Vision Color Mixing versus Pigment Mixing 
Colorimeters Color Matching Graphical Representations of 
Chromaticity. 

EXPERIMENTS IN LIGHT 

1. FOCAL LENGTHS OF SIMPLE LENSES 343 

2. CARDINAL POINTS OF LENS SYSTEMS 347 

3. A STUDY OF ABERRATIONS 349 

4. MEASUREMENT OF INDEX OF REFRACTION BY MEANS OF A 

MICROSCOPE 352 

5. THE PRISM SPECTROMETER 353 

6. THE SPECTROPHOTOMETER 358 

7. INDEX OF REFRACTION BY TOTAL REFLECTION 365 

8. WAVE-LENGTH DETERMINATION BY MEANS OF FRESNEL'S BIPRISM. 368 

9. MEASUREMENT OF DISTANCE WITH THE MICHELSON INTERFEROM- 

ETER 370 

10. MEASUREMENT OF INDEX OF REFRACTION WITH A MICHELSON 

INTERFEROMETER 376 

11. RATIO OF Two WAVE-LENGTHS WITH A MICHELSON INTERFEROM- """ 

ETER 380 

12. THE FABRY-PEROT INTERFEROMETER 382 

13X MEASUREMENT OF WAVE-LENGTH BY DIFFRACTION AT A SINGLE 

SLIT 384 

14T THE DOUBLE-SLIT INTERFEROMETER 387 

15. THE DIFFRACTION GRATING 390 

16. SIMPLE POLARIZATION EXPERIMENTS 395 

17. ANALYSIS OF ELLIPTICALLY POLARIZED LIGHT WITH A QUARTER- 

WAVE PLATE 399 



CONTENTS xi 

PAOK 

THE BABINET CQMPENSATOK 401 

ROTATORY POLARIZATION OF COMMON SUBSTANCES 403 

20. VERIFICATION OF BREWSTER'S LAW 407 

21. THE OPTICAL CONSTANTS OF METALS 410 

22. POLARIZATION OF SCATTERED LIGHT 412 

23. THE FARADAY EFFECT 414 

APPENDICES 

I. A COLLINEAR RELATION USEFUL IN GEOMETRICAL OPTICS . . . 419 
II. THIRD-ORDER CORRECTION FOR SPHERICAL ABERRATION FOR A 

THIN LENS IN AIR 421 

III. DERIVATION OF EQUATIONS FOR ASTIGMATIC FOCAL DISTANCES 

AT A SINGLE REFRACTING SURFACE 424 

IV. ADJUSTMENT OF A SPECTROMETER 426. 

V. PREPARATION OF MIRROR SURFACES 430 

VI. MAKING CROSS HAIRS 435 

VII. STANDARD SOURCES FOR COLORIMETRY 436 

VIII. THE FRESNEL INTEGRALS 438 

TABLES OF DATA 

I. USEFUL WAVE-LENGTHS 443 

II. INDICES OF REFRACTION OF SOME COMMON SUBSTANCES .... 444 

III. REFLECTING POWERS OF SOME METALS 445 

IV. FOUR-PLACE LOGARITHMS . 446 

V. TRIGONOMETRIC FUNCTIONS 448 

VI. LOGARITHMS OF TRIGONOMETRIC FUNCTIONS 452 

INDEX 461 



LIGHT: PRINCIPLES AND 
EXPERIMENTS 

CHAPTER I 
FUNDAMENTAL CONCEPTS IN GEOMETRICAL OPTICS 

1. Fundamental Postulates. Optical phenomena may be 
divided into two classes. The most important of these in the 
light of modern experimental discovery is that which is included 
in the subject of physical optics, which deals with theories of the 
nature of light and of its interaction with material objects, 
together with experimental verification of these theories. Funda- 
mental to the study of physical optics, however, is a knowledge 
of the principles of another class of optical phenomena which, 
after the introduction of a few fundamental experimental facts, 
may be described without taking into account any hypotheses 
concerning the nature of light or its interaction with material 
bodies. This division of optics, concerned with image formation 
by optical systems and with the laws of photometry, is called 
geometrical optics, since its description is founded almost entirely 
on geometrical relations. Because an understanding of the laws 
of image formation is fundamental, geometrical optics will be 
dealt with first. 

There are certain experimental facts, sometimes regarded as 
postulates, upon which the study of geometrical optics may be 
based: 

1. Light is propagated in straight lines in a homogeneous medium. 

2. Two independent beams of light may intersect each other and 
thereafter be propagated as independent beams. 

3. The angle of incidence of light upon a reflecting surface is equal to 
the angle of reflection. 

4. On refraction, the ratio of the sine of the angle of incidence to the 
sine of the angle of refraction is a constant depending only on the nature 
of the media (Snell's law). 

1 



2 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. I 

To these four facts may be added the concept of the ray and 
certain deduced laws which are subject to experimental 
verification. 

2. The Ray. The ray may be defined as the path along which 
light travels. Since for most purposes it is possible to consider 
the light to be a wave motion spreading out with the same velocity 
in all directions from the source (in a homogeneous and isotropic 1 
medium), we may say that the ray is the direction in which this 
wave motion is propagated. Indeed, it is not necessary to 
specify the wave form of the light, but simply to consider it to 
be propagated in straight lines, since any consideration of the 
physical nature of the light takes us outside the realm of geo- 
metrical optics. While some exception may be taken to the use 
of the ray concept as not conforming to modern ideas of the 
nature of light, it is found most convenient in discussing the 
characteristics of optical systems to trace the paths of the rays 
from a source through succeeding media in accordance with the 
preceding four laws. 

3. The Optical Length of a Ray. It has been proved experi- 
mentally that light undergoes a change in velocity in passing 
from one medium to another, and that the index of refraction 
given by Snell's law, n = sin ^'/sin r, is also given by 

_ velocity in vacuo 
~ velocity in the medium 

As given here, n & is the absolute index of the medium. Since 
the velocity of light in air is very little different from that in 
vacuo, for optical purposes the index of air is taken as unity. 
For example, the index of refraction of glass is commonly given by 

__ velocity in air 
~~ velocity in glass' 

this is the ratio of the absolute index of glass to that of air. 

The optical length of a ray of length I'm a medium of index n 
is denned as the product nl. Light rays from a point source at 

1 A medium is said to be optically isotropic when it has the same optical 
properties in all directions. Thus, water, and glass free from strains, are 
isotropic. Glass with strains, and all crystals except cubic, are anisotropic. 
On the other hand, any one of these is homogeneous if different portions of its 
mass have the same characteristics. 



SBC. 1-4] 



FUNDAMENTAL CONCEPTS 



on the optical axis of a lens (Fig. 1-1) reach the lens at its ver- 
tex B sooner than at any other point, A. At the surfaces the 
rays will undergo refraction and, if the lens is free from aberra- 
tions, will converge to an image point /. If the distance BB' 
is greater than AA' the retardation along the axis in the glass 
will be more than between A and A'. While the linear path 
OAA'I is greater than OBB'I, the optical paths are the same; i.e., 
the times taken by the light to go from to / over the two paths 
are the same. 

Let the indices of refraction of air and glass be n a and n a , 
respectively. Then the optical paths 

OA - n n + AA' - n a +A'I - n a and OB n a + BB' - n a + B'l - n a 



are the same. A more general statement is that 2^ / is 
constant for all rays traversing a perfect optical system, where k 




FIG. 1-1. 

is the linear distance in each medium of index of refraction n. 
In ordinary lens systems the statement would be true only for 
two adjacent rays. 

4. Fermat's Principle. If, in Fig. 1-1, the angle made by 
the ray OA with the axis is 6, then 

I) 



BO ' 

This is the mathematical statement of a principle first stated by 
Fermat, the principle of least time, which says that the path taken 
by light in passing between two points is that which it will 
traverse in the least time. 

Sometimes the general law expressed by Fermat's principle is 
called the law of extreme path. Light reflected from a plane sur- 
face at P, in Fig. 1-2, travels from A to B by the shortest path 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. I 




A' 



s 



Fio. 1-2. 



APB. To prove this, consider the distance of the virtual image 
A! from B through P as compared to the distance through any 
other point P' on the surface. According to the law of reflection, 
i = i', hence APB is the actual path of the light, and is equal to 
A'PB, which is shorter than any other path A'P'B. In this case 

the "extreme" path is the short- 
est path; in other cases, however, 
"extreme" may mean either a 
maximum or a minimum. 

Illustration may be simplified 
by introducing the aplanatic sur- 
face. A reflecting or refracting 
surface is aplanatic if it causes 
all rays incident upon it from an 
object to converge to a single 
image oint. Thus, an ellipsoid 
of revolution is an aplanatic surface by reflection for a point object 
placed at one focus, the image point being the other focus, since 
the sum of the distances from the two foci of the ellipsoid to 
any point on the surface is constant. 

Ah aplanatic refracting surface is illustrated in Fig. 1-3 by the 
curve SPS'. The equation of such a surface is 

n\ - AP + n 2 - PB = constant, 

where n\ and n z are the indices of refraction of the two media 

and AP and PB are the linear 

distances, respectively, from 

the object point to the surface, 

and from the surface to the 

image point. The surface is 

concave toward the medium of 

greater index, n 2 ; consequently 

the optical path 

wi AQ + n z QB Fw. 1-3. 

is the same as that through the point P. 

Now suppose the rays to be refracted, instead of at the surface 
SPS', at another surface, through P and Qi, of greater curvature 
than SPS', and tangent to the first surface at P. Then 




SEC. 1-6] FUNDAMENTAL CONCEPTS 

m AP 4- w 2 PB = m AQ + n 2 Q5 

= i AQ -f n 2 QQi 4- ^2 Q\B 

> ni - AQ + ni QQi + n 2 Qi 

(since n 

> ni AQi + n 2 QiB 

(since rii AQ -f wi QQi > HI 



Since the point Qi is any point on the second surface except P, 
the optical path of the light through P is a maximum for this 
surface. 

On the other hand, consider the light to be refracted from a 
third surface, passing through P and Qz, but of smaller curvature 
than SPS'. By an argument similar to the preceding one, the 
optical path of the ray refracted at P can be shown to be less than 
that of any other ray refracted at the third surface, and hence to 
be a minimum. 

Thus the optical path of a ray by refraction may be either a 
maximum or a minimum. 

6. The Principle of Reversibility. By referring to Fig. 1-1 
it will be seen also that a ray starting from 7 and traversing the 
path I A' must of necessity be subject to refraction through the 
lens which will make the ultimate path of the ray AO. The 
fact that the direction in which the light is propagated may be 
reversed without changing the path of a ray is known as the 
principle of reversibility. 

6. The Law of Malus. From the geometrical laws already 
stated, particularly from Fermat's principle, may be deduced 
another principle, the law g of Malus, which states that an ortho- 
tomic system of rays remains orthotomic after any number of 
refractions and reflections. An orthotomic system is one which 
contains only rays which may be cut at right angles by a properly 
constructed surface. The geometrical proof will not be given 
here. It is evident that if we consider the light to be radiated 
from a point source in all directions, the surface of a sphere 
about the point will, in a homogeneous and isotropic medium, 
constitute the surface cutting the rays at right angles. The 
passage of the light into another medium will give rise to another 
surface which, although not a sphere having its center at the 
source, will nevertheless cut all the rays at right angles. An 
extended source may be considered as a multiplicity of point 



6 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. I 

sources. From the standpoint of the wave theory, in which we 
may regard the ray as the direction of propagation of the wave, 
the law of Malus needs no proof. 

7. The Focal Length of a Thin Lens. A "thin" lens is one 
whose thickness is negligible compared to its focal length. 

In a simple thin lens, the optical axis is the line through the 
center of the lens joining the centers of curvature of the surfaces. 
If the lens is used to form an image of an object, then the rela- 
tion 



a a' f 

holds, when a, the distance from object to lens, a', the distance 
from image to lens, and /, the principal focal length of the lens, 
are measured along the optical axis. It will be shown in the 
following chapters that I//, sometimes called the power of the 
lens, depends only on the radii of curvature r\ and r 2 of the sur- 
faces and the index of refraction n of the substance used, and is 
given by 

1 




If in eq. 1-1 a is put equal to infinity, a' = /. By definition, the 
focal length of a simple thin lens is the distance from the lens at 
which all incident rays parallel to the axis will meet after refrac- 
tion. Similarly, if a' = , a = /; the lens thus possessing two 
principal focal points. 

8. Two Thin Lenses. If two thin lenses are used coaxially, 
the focal length / of the combination depends upon their focal 
lengths /i and /? and the distance d between them and is given by 

1 1 1 d 

-f = 7- + 7 - 
/ fi h 



This relationship will be developed in the following chapters. 

9. The Concept of Principal Planes. It is evident that the 
distance / in eq. 1-3 is not in general measured to any of the four 
surfaces of the lenses. Nevertheless, the principal focal length 
must be measured to some axial point. Only in the simplest 
cases of single thin lenses, or of combinations of thin lenses very 
close together is the principal focal length given even approxi- 
mately by the distance to the lens from the point where incident 
parallel rays meet. For thick lenses and most combinations it is 



SEC. 2-5J 



THE LAWS OF IMAGE FORMATION 



11 



the image space there is a point /i, with coordinates (x' t y'), 
conjugate to Oi. The point F is the principal focal point in the 
object space. If a point source of light is placed at F, all the 
rays which are emergent from the optical system will be parallel 
to the optical axis XX'. Similarly, the point F' is the principal 
focal point in the image space. Rays which are parallel to the 
optical axis in the object space will, after interception by the 



(0.0) 



(0.0) 



F' 



Fi. 2-2. The coordinates in the object and image spaces. 

optical system, meet at F 1 . In the figure y' is negative, illus- 
trating the case for a real image formed by an ordinary 
double-convex lens. The rays proceed from left to right. By 
convention, distances in the object space are positive to the 
right of F, and in the image space to the left of F'. 

4. Lateral Magnification. The ratio y'/y in eq. 2-7 is known 
as the lateral magnification and is characterized by the symbol 0. 



n' 



I- a :- *K~~~ '- \ 




4 



5. Collinear Equations for a Single Refracting Surface. If the 
system is a single refracting surface, then, in eq. 2-1, a = / x, 
and a' = /' x f . Substituting these values in eq. 2-5, we 
obtain xx' ff r , which is eq. 2-6. To obtain eq. 2-7 for a single 
surface we may proceed as follows: Consider an object 00\ and 
its conjugate image //i, as illustrated in Fig. 2-3. Putting 



12 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. II 



= y and II\ = y', and assuming that y and y f are small 
compared to 0V and 7F, we may write 

tan y _ y(f' x'} .__ sin ^ _ n' 
tan ^>' ?/'(/ x) ~ sin <p' ~~ n 

From this it follows, using eq. 2-4, that 



2/'(/ - *) / 

which by simplification becomes y' /y = f/x = x'/f, which is 
eq. 2-7. 

6. Principal Points and Planes. It should be pointed out that 
the distances / and /' as obtained from the collinear relation are 
not necessarily the focal lengths in the object and image spaces; 
thus far this has only been shown to be true for a single refracting 



> 






F' 



FIG. 2-4. The principal (unit) planes are where x = f and x' f. 

surface. In coaxial systems in general they are thus far con- 
sidered only as two numbers whose values depend upon the 
characteristics of the optical system, such as the radii of curva- 
ture of the surfaces, the indices of refraction of the media, and 
the distances between the surfaces. They can be given a more 
definite meaning for ideal systems by considering eq. 2-7. The 
value of the lateral magnification, 0, will be unity when / = x 
or when /' = x'. Since x and x' are the distances from the 
principal focal points to the object and image planes, respectively, 
the value = 1 defines two planes perpendicular to the optical 
axis whose distances from F and F' are f ( x) and f (~x r ). 
These planes are illustrated in Fig. 2-4 by the lines marked P 
and P' perpendicular to the optical axis. These planes are called 
the unit or principal planes. Their intersections with the axis 
are called the principal points. By eqs. 2-6 and 2-7, for these 
values of x and x', y' = y, and both are on the same side of the 
axis. Moreover, nothing in the development of the collinear 



SBC. 2-7] 



THE LAWS OF IMAGE FORMATION 



13 



eqs. 2-6 and 2-7 requires that the principal planes be located 
between the focal points F and F' as shown in Fig. 2-4, but only 
that the distances from F to P and from F' to P' have the same 
sign for the condition 1. 

7. Conjugate Rays and Conjugate Points. Although the 
concept of conjugate points has been introduced in Sec. 2-3, 
some further discussion of it is worth while. As a result of the 
one to one relation existing between points, lines, or planes in the 
object space and image space, it follows that corresponding to 
every ray originating at an object point and lying in the object 
space there is a second ray in the image space which is a con- 
tinuation of the first. These two rays constitute a pair of 
conjugate rays. Moreover, corresponding to each point lying 
on a ray in the object space there is a point lying on the conjugate 
ray in the image space. Any such two points constitute a pair of 
conjugate points. In Fig. 2-3, and / are conjugate points, as 
are also 0\ and I\. Similarly 0\V and I\V are conjugate rays. 




Fio. 2-5. Illustrating conjugate rays and points. Ii is conjugate to Oil Iz is 
conjugate to Of, the conjugate to F is at infinity; A' is conjugate to A. 

The principal planes of an optical system have the important 
property that a pair of conjugate rays will intersect the planes at 
equal distances from the axis. The realization of this will be 
easier if it is considered that y and y' need not necessarily be 
the distances of points 0\ and /i from the axis, but may be the 
distances from the axis of another pair of points, provided these 
points are also conjugate one to the other. For instance, in 
Fig. 2-5, the ray 0\FA must emerge from the system at A', 
since P and P' are defined as a pair of planes for which / x and 
f = x'. This ray, moreover, must, after leaving A', proceed 
parallel to the axis, since in the object space it passes through 



14 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. II 

F. If the object point were any other point on the line 0\FA 
except Oi, this would still be true. For any other point, such 
as Oz, however, the conjugate point in the image space would not 
be at /i but at some point such as I 2 . Similarly, there will be a 
ray Ai'F'Ii conjugate to the ray 0\A\, and a ray A 2 '/i conjugate 
to the ray 0\A^. But for all such pairs of conjugate rays, there 
is only one pair of planes for which (3 = 1 and these are the 
principal planes of the system. In Fig. 2-5 we may see also that 
the distances / and /' of these planes from the principal focal 
points F and F f may be regarded as the principal focal lengths 
of the system. A comparison with the definitions of / and /' 
given in eqs. 2-2 and 2-3 shows that the principal planes of a 
single refracting surface coincide and cut the axis at the vertex 
of the surface. It is also evident that only in the case where the 
indices of the initial and final media are the same will / = /'. 

8. LaGrange's Law. Returning to a further consideration 
of Fig. 2-3, it follows that since /' x' = a' and / x a, the 
equation for the lateral magnification may be written 

ft = y - = ^ (2-8) 

y 



provided the angles <f> and <?' are small. If we consider in addi- 
tion a paraxial ray, i.e., one which makes a very small angle with 
the axis and lies close to the axis throughout its length, from 
to 7, then, putting A V h, we have 

h = au = a'u', (2-9) 

4 

in which u and u' are the angles made by the ray in the object 
and image spaces, respectively. Also, for small angles, SnelPs 
law may be written 

^ = 1. (2-10) 

<p n 

Combining eqs. 2-8 and 2-9, there results 

(2-11) 



y 
and from eq. 2-10 it follows that 

nyu = n'y'u', (2-12) 



SEC. 2-10] 



THE LAWS OF IMAGE FORMATION 



15 



which is known as LaGrange's law, and sometimes as the Smith- 
Helmholtz law. It may be shown that this law can be extended 
to the case of refraction at any number of successive surfaces, 
provided y and u are both very small. This is tantamount to an 
assumption that the rays under consideration are paraxial rays. 

9. Longitudinal Magnification. From elementary considera- 
tions, it is evident that for an object of any depth along the a>direc- 
tion there will be a corresponding depth in the image. Indicating 
these distances by da and da', respectively, we may define the 
longitudinal magnification a as the ratio da' /da. By differentia- 
tion of eq. 2-5 it follows that 



a 



"2 



a = 



/ 



da 



(2-13) 



10. Angular Magnification. Nodal Points. Consider a ray 
from some point 0\, not on the axis, to intersect the axis at a 




FIG. 2-6. 

point E, as in Fig. 2-6, and tho incident principal plane at A. 
There will be a ray conjugate to this emerging from A' and inter- 
secting the axis at some point E' '. It is evident that the axis 
constitutes another pair of conjugate rays passing through E 
and E'. Hence a point object at E will give rise to a point image 
at E f . If the angles made by EA and E'A' with the axis are 
u and u', respectively, then the angular magnification y may be 
represented by 7 = tan w'/tan u. But this is equal to a/a', 
since y = y'. We have, however, established for all ideal optical 
systems the identity of / and /' with the focal lengths in the 
object and image spaces. In consequence, it follows that a/a' = 
(/ x)/(f x f ), and from eqs. 2-6 and 2-7 we have finally that 



_ _ - _ 
7 ~ x 7 ~ f 



(2-14) 



16 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. II 



When the angular magnification, 7, is equal to 1, /' = x and 
/ = x'\ also tan u 1 tan u. In this case the conjugate rays 
are parallel and intersect the axis at two points N and N f called 
the nodal points of the system, as shown in Fig. 2-7. 

The focal points F and F f , the principal points P and P', and 
the nodal points N and N f are called the cardinal points of an 
optical system. From the character of their definitions they 
give a description of the system and its effect on the rays incident 
upon it. 

P P r 



FIG. 2-7. The nodal points (NN r ) are the (conjugate) intersections with the 

axis of a pair of conjugate parallel rays. 



Disregarding the nep^tive sign on the right-hand side of eq. 
2-13, combining PT> 2-7, 2-13, and 2-14 results in 

j3 = 7 a. (2-15) 

Also, since y = a/a', it follows from eqs. 2-8, and 2-10 that 

<p' n 

/3 7 = = 
(p n 

By adopting the convention that in the case of a real image u 
and u' are of the same sign, while y and y' are of opposite sign, ' 
we obtain 







hence 



f _n 



(2-16) 



11. Mirror Systems. The equations and concepts which 
have been developed in the preceding paragraphs for refracting 



SBC. 2-11] THE LAWS OF IMAGE FORMATION 17 

surfaces can be used with slight modifications for mirrors. In 
Fig. 2-8, 

. _ (a - r) sin p (r - a') sin p 
sin * ----- g ----- = - p - , 

from which it follows that 

(a r) _ (r a') 
b P 

For paraxial rays, b = a and b' = a' approximately, so that 

I + J- = ?. (2-17) 
a a' r 

This is analogous to eq. 2-1. Since 
for small angles r = 2/, it follows 
that for a mirror j "j j-a'~j5 

k -------------- a -I ------ ......... \ 

111 k ----- r ------- 1 

- + -3-- (2-18) 




The conventions already adopted may be used for the case of 
mirrors also. In Fig. 2-8, r is negative, while in the case of a 
convex mirror, r would be positive. 

Problems 

1. Given a lens system for which /= -MO, /' = +8, x = 12, 
y +6. Using a diagram, find x' and y'. 

2. Given an optical system for which / = +10, /' = 16, x = 20, 
y = 0. Using a diagram, find x'. 

3. How far from a convergent mirror must an object be placed to 
give an image four times as large, if the focal length of the mirror is 
50 cm.? 

4. An object is 1 m. in front of a concave mirror whose radius of 
curvature is 30 cm. It is then required to move the image 15 cm. 
farther from the mirror. Through what distance must the object be 
moved, and which way? 

5. An object is placed between two plane mirrors which are inclined 
at an angle of 60 deg. How many images are formed? 

6. What must be the angle between two plane mirrors if a ray inci- 
dent on one and parallel to the other becomes after two reflections 
parallel to the first? 

7. A small bubble in a sphere of glass 5 cm. in diameter appears, 
when looked at along the radius of the sphere to be 1.25 cm. from the 



18 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP, II 

surface nearer the eye. What is its actual position? If the image of 
the bubble is 1 mm. in height, what is its real diameter? What will be 
the longitudinal magnification? (Assume n = 1.5) 

8. A spherical bowl of liquid has a radius of 10 cm. For what index 
of refraction will the focus of the sun's rays be at one side, i.e., at P^Pa 7 ? 

9. A spherical bowl of 20 cm. radius is filled with water. What will 
be the apparent position of a bubble, seen along a radius, which is 
15 cm. from the side of the bowl? What will be the lateral magnifica- 
tion ? The longitudinal magnification ? 

10. What must be the focal length of a lens which will give an image 
of the sun 6 in. across? 

11. Derive the expression for the longitudinal magnification a from 
eq. 2-6, and show that it is the same as given in eq. 2-13. 

12. An object lies 250 mm. in front of the incident nodal point of a 
lens whose focal length is +60 mm. Where is the image with respect 
to the emergent nodal point? Use a diagram in answering the question. 



CHAPTER III 

COMBINATIONS OF OPTICAL SYSTEMS 

1. Equation for a Thin Lens. In Sec. 2-2, by considering the 
refraction of rays at a spherical surface, it was found that the 
distance a' of an image point on the axis from the vertex of the 
surface was related to the distance a of the conjugate object point 
from the same vertex by eq. 2-1 : 



n . n' n' n 



a a r 

in which n is the index of refraction of the medium in the object 
space to the left of the surface r, and n' is the index of the medium 
of the image space to the right of the surface. Equation 2-1 is 
based upon the important hypothesis that the aperture of the 
optical system, in this case consisting of a single refracting sur- 
face, is small compared to the other dimensions involved. Two 
rays were considered, one constituting the optical axis, the other 
a paraxial ray OAI (Fig. 2-1) incident upon the surface at a 
relatively short distance from the axis. To continue this pro- 
cedure and thus derive a lens formula for an ideal system of more 
than one surface, with a distance of any appreciable amount 
between the surfaces, would be extremely cumbersome. It is 
relatively easy, however, to obtain the formula for a thin lens. 
As the term is used here, a thin lens means one in which the 
distance between the surfaces is so small relative to other dimen- 
sions that it may be ignored. 

In Fig. 3-1, the essential features of Fig. 2-1 are reproduced. 
The radius of curvature of the first surface is now called r\ 
and there is added a second surface of radius r 2 . Both n and r 2 
are by convention positive, and the medium to the right of the 
second surface has the index n". As in eq. 2-1, the image dis- 
tance obtained by refraction at the first surface only is 

n n' ^_ n' n C*-\\ 

a a m ' ~ n 
19 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. Ill 



where a m f is used for the image distance, to distinguish it from 
a', which will be reserved for the image distance for the entire 
lens. 

With regard to the second surface, the conjugate points I m and 
/ have the relation of object and image. Hence we may write 
an equation analogous to eq. 3-1, 



a 



m 



a 



n" - n' 



(3-2) 



the object distance o m ' for the second surface being negative. 
Adding eqs. 3-2 and 3-1, we obtain 



n 



n' n n" n' 



(3-3) 



a a r\ r% 

If the system is a thin lens in air, n = n" = 1, and n' may be 




o 



Iff* 



FIG. 3-1. 



called n, the index of refraction of the glass, whereupon^eq. 3-3 
becomes 



. ( 

- -f = (n 

a a 



TI r 2 



(3-4) 



Since by definition the principal focus of a system is that point at 
which incident rays parallel to the axis will meet, by the substi- 
tution of oo for a in eq. 3-4, a' becomes /, the focal length of the 
lens, and the right-hand member of this equation is equal to 

i//. 

By comparison with eqs. 2-5 and 2-18 it will be seen that the 
focal length for any system in air is given by 



SBC. 3-2] COMBINATIONS OF OPTICAL SYSTEMS 



21 



In using eq. 3-4 it is important to remember that r\ and r 2 are 
positive when the surfaces are convex toward the object. For a 
surface concave toward the object, the sign of r must be changed. 
2. Combinations of Two Systems. Since the equations 
developed in Chap. II apply to any ideal optical system, i.e., 
one in which the sizes of the apertures and objects are limited, 
they can be used for an ideal system composed of two coaxial 
parts. These parts may consist of separate lenses placed 
coaxially, of lens and mirror combinations, or of several refracting 
surfaces placed coaxially so as to constitute an image-forming 
system. It is the purpose here to show how the cardinal points 
and equations for the focal length of the combination can be 
expressed in terms of the characteristics of the separate parts. 




FIG. 3-2. 

In Fig. 3-2 is shown a ray passing through two systems having 
a common axis. The subscript 1 refers to the first system, the 
subscript 2 to the second, and symbols with no subscript to 
the combination considered as a single system. As before, 
primed symbols refer to the image spaces for the -systems, and 
unprimed symbols to the object spaces. In accordance with the 
procedure in Sees. 2-3 to 2-10, inclusive, the origins of the systems 
will be the focal points. For example, the point FI is the origin 
in the object space in the first system, FI is the origin in the 
conjugate image space,- and F r is the origin in the image space for 
the combination. 'The ray incident to the entire system is 
parallel to the optic axis and will consequently pass through 
F\ and F'. Let hi = hi represent the distance from the axis 
of the intersections of the ray with Pi and Pi, and hj = h^ 
represent the distance from the axis of its intersections with /Y 
and P 2 . Let A, the separation of the principal focal points FI 
and F z , be positive when there is no overlapping of the inner focal 



22 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. Ill 



distances F\P\ and FzPz as shown in the figure, and negative 
when overlapping exists. Then the angular magnification 72 of 
the second system is given by 

_ 2 _ A 

72 f i ft' 

h h 

since Xz is, by convention, negative to the left of F* But since 
u' = W, and w 2 = u\, by eq. 2-14 

tan u% tan 

72 = 



tan 7*2 tan u\ hi/f\ f 

as h f for the entire system is equal to h\, because the ray must 
cross P' for the entire system at the same distance from the axis 
at which it is incident upon P. The negative sign is used for /' 
since the principal focus F' lies to the left of P'. (If A is negative 
for a combination of two lenses, i.e., if the focal distances f\ and 
/ 2 overlap, then /' will be positive.) Hence 



f = JlK, and similarly, / = -^ 2 - (3-5) 
By the use of eq. 2-6 it is also possible to show that 

, and similarly, F,F = -'- (3-6) 

The distance p'^PTY) = ft + Ft'F' +f, hence from eqs. 3-5 
and 3-6, 



A 

+ /iA 



/Y\ - -^^ V -T 

P - A ' % 

and since d = fi + A -f / 2 , these can be reduced to 



d-fS - 






fid 



(3-7) 



It is further evident that consideration of a ray passed through 
the system in the opposite direction will yield all the necessary 
relations in the object space. 



SEC. 3-3] COMBINATIONS OF OPTICAL SYSTEMS 23 

For a combination in air of two lenses of focal lengths /i and/2, 



/- ? 

J A d - fi - /,' 



- + - - {3 ' 8) 



or 



3. A General Lens Formula. -Methods have been described for 
obtaining the characteristics of image formation by refracting sur- 
faces, and it has been shown that the fundamental formulas of 
ideal lens systems may be obtained by applying the principles 
of projective geometry to the optical case. Often it is found 
desirable to introduce the concept of the power of a system in 
increasing the convergence of the rays incident upon it. A lens 
in air is said to have a power of 1 diopter when its focal length is 
1 m.; one having a power of 10 diopters has a focal length of 
0.1 m. Thus the power (P of a lens in air is the reciprocal of its 
focal length in meters. 

In a more general case, however, the index of refraction of the 
medium into which the rays emerge must be considered. For 
example, if light is incident in air upon a lens sealed to the end of 
a tube of water, the focal length /' in the water will be greater 
than the focal length / in air. A more extreme case would be 
that of a lens immersed in a medium of higher index than that of 
the glass. In this case the lens, convergent in air, would be 
divergent in the medium of higher index. In a divergent system, 
i.e., one which decreases the convergence of the rays incident 
upon it, the power is a negative quantity. 

Using the concept 6f power of convergence described above, 
a general lens formula may be obtained. 1 In Fig. 3-3 the shaded 
area bounded oA trfet, right by the surface Si represents a system 
upon which light is incident from the left. Let y be the distance 
from the axis of a ray parallel to it, and let h be the distance 
from the axis at which the ray leaves Si. If the surface S 2 were 
not present, such rays parallel to the axis would converge to 
FQ', and the focal length of the system A to the left of Si would 
be /o'. The addition of S 2 , cut by the ray under consideration at 
a distance h from the axis, will cause the ray to cross the optical 

1 The elegant method here described was originated by Professor C. W. 
Woodworth. 



24 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. Ill 



axis at F', and the focal length of the combination will be/ 7 . The 
value of fo will depend upon the index of refraction of the medium 
between the surfaces Si and $2, and the value of /' upon that to 
the right of Sz. Hence we may redefine the power of the system 
as the index of refraction divided by the focal length; i.e., 
(Po = n/fj and (Pi = n'/f 1 . Assuming that the aperture is so 
small that ho and h may be considered to lie in the surfaces Si 




and $2, we get, from similar triangles, to a sufficient degree of 
approximation, 



h . f i _ h . n 
-'/o -- 



(3-9) 



and 



= - - -. (3-10) 

n i t\J 

if vj VJ 

But by eq. 2-1 the object and image distances for a single refract- 
ing surface are given by 

n' n 



_ 
' 



a a 



In the present case, a' V>J<", and a 
3-9 and 3-10 



, hence from eqs. 



(3-11) 



, kin' - n\ 
(P = (P + -I ) 

y\ r I 

The second term on the right-hand side gives the amount by 
which the power of the system will be changed by the addition 



SBC. 3-3] COMBINATIONS OF OPTICAL SYSTEMS 



25 



of a refracting surface of radius r. There will be a similar term 
for every such surface added, hence eq. 3-11 is a recurrent 
formula, and for any system may be written 



(Pi - 



h\(ni - n_A 

y)\rT~')' 



(3-12) 



The value of the h at each added surface may be obtained as 
follows: In Fig. 3-3 



t = 



V iF ' - 



where KQ refers to the distance from the axis at which the rays 




FIG. 3-4. 



emerge from the system A. Substituting in this the value of 
/</ from <P = n// ', 

A- -sT\ 

(3-13) 



n 



If y is put equal to unity, eq. 3-11 can be simplified to 

(Pi = (Pi-i + -(wi - rii- 



(3-14) 



and, using the general subscript i as before, eq. 3-13 becomes 



hi = ftt-i 



n 



(3-15) 



in which n is the index for the part of the system in which t lies. 
The equation for a single lens may now be found. In Fig. 3-4 
a lens of index n in air has surfaces of radii n and r 2 , and a thick- 
ness between its vertices of t. The power of the first surface is 
given by eq. 3-14 






(3-16) 



26 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. Ill 

ft 

since for parallel light entering the lens (Po = 0, and h\ = y\ = 1. 
For the second surface, by eq. 3-15, h z hi -- J > or, substi- 

71 

tuting the value of (Pi from eq. 3-16, 



n ri 
Also, by eq. 3-14, (P 2 = (Pj H - 2 (1 ri), which on substitution 

7*2 

of /i2 from eq. 3-17 and (Pi from eq. 3-16 becomes 

n - 1 n - 1 . t (n - I) 2 

<P = ---- 1 --- ^ - '-- 
r* r2 n 



This is the power of the entire lens, which may be written 

<P C = (n - 1)(- - - + - i-Z_l) = i (3-18) 
\ri r 2 n r^ / f 



For a thin lens in air, t may be put equal to zero, and eq. 3-18 
is reduced to the familiar form 

= (n - l)i - t. (3-19) 



It is frequently desirable to know the distance from the back 
face of the lens to the emergent focal point F'. This is given by 
the ratio h z /hi = ///', from which, since hi = 1, 



' 



= 2 = - . -Z. (3-20) 



By means of eq. 3-18 v' can also be expressed in terms of r z 
instead of r\. 

By methods similar to that above, the equation analogous to 
eq. 3-18 for I//, and one analogous to eq. 3-20 for v, may be found. 
If the lens system is in air, / = /'. It is evident that, in order 
to obtain the focal length of a system, eqs. 3-14 and 3-15 may be 
used successively for as many surfaces as there are in the 
combination. 

4. Classification of Optical Systems. Often a lens or mirror 
is designated as convex or concave, according to the shape of its 
surface. The difficulty in this usage is that simply the concavity 



SEC. 3-5] COMBINATIONS OF OPTICAL SYSTEMS 27 

or convexity of the surfaces is not enough to describe the character 
of the system. A more useful procedure is to describe a system 
by its effect upon the light incident on it, i.e., the convergence or 
divergence imposed upon the rays. 

Convergent systems can be characterized as dioptric or katop- 
tric. The former are those in which the image moves to the right 
as the object moves to the right, i.e., toward the lens system, 
while the latter are those in which the image moves to the left 
as the object moves to the right. Thus it will be seen that a 
"double-convex" lens, of index greater than unity, is convergent 
and dioptric, since no matter where the object is, as it moves to 
the right the image does likewise. On the other hand, a concave 
mirror, also convergent, is katoptric since the image moves to the 
left as the object moves to the right. A combination of two 
such mirrors is dioptric. Hence there is a general rule that a 
dioptric system is one composed of one or more refracting sur- 
faces, or these combined with an even number of reflections, 
while a katoptric system is composed of an odd number of 
reflections, or combinations of these with refractions. Similarly, 
divergent systems may also be characterized as dioptric or 
katoptric. 

Since the difference produced in a lens by changing from convex 
to concave refracting surfaces is a difference in the signs of the 
principal foci, we can classify optical systems as follows: 

Convergent: Dioptric :/ positive, /' positive 

Katoptric : / positive, /' negative 
Divergent: Dioptric : / negative, /' negative 

Katoptric -./negative,/' positive 

If a lens system is classified according to its power of increasing 
the convergence of the rays incident upon it, a convergent system 
is said to-be positive, while a divergent lens is negative. A positive 
lens may also be defined as one which forms an inverted image 
of a distant object. 

A simple lens which has a greater thickness between its 
vertices than at its rim is convergent, arid one which is thinner is 

divergent. 

5. Telescopic Systems. In the strict sense of the word a 
telescope is a combination of two or more lenses, mirrors, or both, 
for the purpose of obtaining magnified images of objects which, 
because of their great distance, appear too small for distant 



28 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. Ill 



vision. The term telescope is also employed, however, when a 
single lens or mirror of great light-gathering power is used to 
enable the observer to photograph images or spectra of distant 
objects, such as celestial bodies. In this case no ocular, or eye- 
piece, is needed. By telescopic systems as discussed in this 
section are meant those combinations of objective and ocular with 
which distant objects are observed visually. When the object 
is very distant, it can be said to be an infinite distance away, 
and the image formed by the objective will be at the emergent 
principal focus. For best vision this point should also be the 
incident principal focus of the ocular, whereupon the rays will be 
parallel upon reaching the eye. Thus we have for consideration 
a coaxial optical system of two parts for which, as shown in 
Fig. 3-5, A = 0. 




FIG. 3-5. The principal planes of a telescopic system. 

For such a system, the equation xx' = ff' has no meaning, 
since x and x' are both infinite, or at least very large compared 
to the other dimensions of the system. Consequently the focal 
distances / and f for the entire system are also infinite or very 
large, and we may choose any pair of conjugate points on the axis 
as origins in the object and image spaces. But although the 
focal positions of object and image may be distant, the relation 
between them is still that of conjugate points. In consequence, 
the ratio between x and x' and the lateral magnification will be 
finite and definite quantities, and we may write 



x' = ax, 



and 



y 



(3-21) 



From the first of these may be obtained by differentiation 
dx' = a dx, which says that the longitudinal magnification a 
of a telescopic system is constant. Since A = 0, i.e., since Fj 
and Fz coincide, 



constant. 



(3-22) 



SEC. 3-5] COMBINATIONS OF OPTICAL SYSTEMS 29 

Also, as A approaches zero, the limiting value of ///' is, by eqs. 
3-6, /i/2//i'/2 ; ; or, for a telescopic system with the same medium 
on both sides, 



r 

Also, the limiting value of a(=x'/x = FiF'/F\F) is, by eq. 3-6, 
given by fzfz/fifi ; or, for a telescope in air, 



a = - (3-23) 

The angular magnification 7 is also constant for a telescopic sys- 
tem. To show this, consider a pair of conjugate rays as shown 
in Fig. 3-6. Let (x,y) and (#',?/') be any pair of conjugate points 
on these rays. Since any pair of points, A and A', on the axis 





FIG. 3-6. 

may serve as origins, the tangents of u and u' are respectively 
y/x and y'/x'. Thus, by eqs. 2-15, 3-22, and 3-23, 

7 - - & (3-24) 

<x jz 

Also, y'u'/yu = |8 2 /a. Since for any optical system this ratio 
is also equal to i> for a telescope in air a = 2 , from which it 

follows that 7 = 1/0, or, the reciprocal of the lateral magnifica- 
tion has the same numerical value as the angular magnification. 
It should be noted that the magnifying power of a telescopic 
system, ordinarily obtained by dividing the principal focal 
length of the objective by that of the ocular, is the angular, and 
not the lateral, magnification. 

Problems 

1. Using diagrams, locate the principal planes of the lenses having 
the following characteristics: 

(a) n = +10, r 2 = -10, t = 2, n = 1.5 
(6) n = -10, r 2 +10, t = 2, n 1.5 



30 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. Ill 

(c) ri = oo, r 2 = + 10, t = 2, n = 1.5 

(d) n = +10, r 2 = oo, t = 1, n = 1.5 
(c) n = +5, r 2 = +10, J = 1.5, n = 1.5 
(/) ri = +10, r 2 = +5, = 1.5, n = 1.5. 
(Note that f is the d of Fig. 3-2) 

2. A sphere of glass has a radius of 10 and an index of 1.5. Using 
a diagram, locate all the cardinal points for the separate refracting 
surfaces and for the whole sphere. 

3. Repeat Prob. 2 above for a hemisphere of glass of the same radius 
and index of refraction. 

4. An air-glass-water system has the following constants: HI = 1, 
7i 2 = 1.5, n 3 = 1.33, r\ = +10, r 2 = 12, t = 2. Using a diagram, 
locate all the cardinal points for the separate components and for the 
whole system. 

5. Using a diagram to scale, locate all the cardinal points of the 
separate components and the whole system for the schematic eye given 
on page 323. 

6. Obtain eq. 3-18 by the relations given in Sec. 3-2. NOTE: make 
use of eqs. 2-2 and 2-3. 

7. A luminous point source is on the axis of a convergent lens, and 
an image is formed 25 cm. from the lens on the other side. If a second 
lens is placed in contact with the first, the image is formed 40 cm. from 
the combination and on the same side as the first image. What is the 
focal length of the second lens? Consider both lenses to be thin. 

8. A bowl of water, spherical in shape, has a radius of 10 cm. Where 
will the focus of the sun's rays be? 

9. What is the focal length of a spherical bubble of air suspended in 
glycerin if the bubble has a diameter of 2 mm.? 

10. What will be the focal length of a sheet of glass bent into cylin- 
drical form, if the thickness of the glass is 2 cm., the index of refraction 
is 1.5, and the radius of the cylinder is 5 m.? 

11. Is it possible to have two thin lenses, one divergent, the other 
convergent, for which / 2 = /i, used together to give an image at a 
finite distance? If so, will the image be real or virtual? Discuss all 
cases, and illustrate them with diagrams. 

12. Using the power formulas of Sec. 3-3, find the focal length of a 
doublet made of a double-convex lens of index n\, and a concavo-plane 
lens of index n 2 , which are in contact. Let r\ = r*, r 3 = r 2 . Call the 
thicknesses of the two lenses ti and < 2 , respectively. 

13. Using the formula derived in the preceding problem, find the 
actual focal length of the achromatic doublet calculated in Sec. 6-16, 
if instead of being a thin lens, the values of t\ and t 3 are 5 and 3 mm., 
respectively. 



CHAPTER IV 

APERTURES IN OPTICAL SYSTEMS 

1. The Stop. If an object is placed before a simple converging 
lens the rays which combine to form the image will be only those 
which pass through the lens. The rim of the lens thus consti- 
tutes the aperture or stop of the optical system. Should the 
image be formed by a simple lens and the eye, it is not certain 
whether the rays which combine to form the image on the retina 
are limited by the rim of the lens or by the iris of the eye. Most 
compound systems, such as photographic objectives, telescopes, 
microscopes, etc., are provided with circular openings which act 
as stops in addition to those which may be due to lens apertures. 
In general an optical system has one stop which is in such a. 
position that it will, by limiting the rays, improve image forma- 
tion as well as provide a restriction on the aperture of the 
instrument. 

The use of stops is not necessarily to reduce the effects of 
faults or aberrations. Even if perfect imagery be assumed, with 
coaxial surfaces as in the ideal optical system, restrictions on 
aperture may be necessary. For the image must be formed on a 
single plane, even if the object has considerable depth. With 
most lens systems, only for points in a given object plane will 
there be sensibly point images in a chosen image plane. Points 
in object planes nearer to, or farther from, the lens will be repre- 
sented by circles of confusion whose dimensions will depend upon 
the longitudinal magnification and upon the size of the cone of 
rays from the object point through the lens system. Limiting 
the extent of this bundle will in general tend to reduce the size 
of the circles of confusion and thus improve the performance of 
the system. 

Another effect of stops in certain positions is to limit the 
extent of the object field for which an image may be obtained. 

2. The Aperture Stop. Consider a simple convergent lens, 
thin enough so that it may be represented by a pair of principal 

31 



32 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. IV 



planes superposed as in Fig. 4-1. Let groups of rays be drawn as 
shown. From the laws of image formation it is evident that 
rays from the object space crossing at E y the edge of the stop S, 
will give a virtual image of E at E'. It will be seen that E' need 



0' 




FIG. 4-1. A front stop as aperture stop. 

not necessarily be between the object and the lens; its position 
along the axis will depend on the character of the image formation 
and the position of E. While S limits the bundle of rays passing 
to the lens from any point on the object, the rays after refraction 
proceed to any point on the image as if limited 1>y stop S 1 . The 
actual stop S is called the aperture stop of the system, and in the 
case described is called a front stop. 

3. Entrance and Exit Pupils. A more general case is that of a 
combination of systems which may be represented by two thin 
convergent lenses as in Fig. 4-2. Here the first lens LI represents 
all the component parts lying on the side of the aperture stop S 
toward 0, and the second lens Lz, all the components on the side 
toward /. Also, LI will give an image of S at some position Si. 
This image is called the entrance pupil. Its position may be 
found by the ordinary laws of image formation. For instance, 
if in Fig. 4-2, is an object position for which S is the aperture 
stop of the system, and L\ a simple lens, then the equation 

* -| = _ gives the position of the entrance pupil. Here a 
a a f 

is the distance from LI to S, a' is the distance from L\ to S\, and 
/ is the focal length of L\. Similarly, there will be at some posi- 
tion S z an image of S produced by L 2 ; this image is called the 
exit pupil If an observer looks through the optical system with 
his eye in the vicinity of 0, he will see the image of S at a posi- 
tion Si, and if he looks through the sysCem with his eye at 7, he 
will see the image of S at a position Si. For an extended object, 



SBC. 4-4] 



APERTURES IN OPTICAL SYSTEMS 



33 



the entrance pupil may be defined as the common base of all 
the cones of rays entering the lens from all the points in the 
extended object. A similar definition may be made of the exit 
pupil. In many cases the position of the aperture stop is not 
restricted to a single place in the system. It might be placed 
anywhere within a considerable range and still be the aperture 
stop. For every such position there will be an entrance pupil 
and an exit pupil corresponding to given object and image posi- 
tions. The stop which performs the duty of aperture stop for 




1*1 



FIG. 4-2. Entrance and exit pupils. 

one position of the object may not do so for another; hence the 
locations of the entrance and exit pupils will depend on the posi- 
tion of the object. In general a good optical system is so con- 
structed that a fixed stop performs the duty of aperture stop for 
object positions over a certain prescribed range. 

Since the entrance and exit pupils are separately conjugate to 
the aperture stop, they are conjugate to each other. For instance, 
in Fig. 4-2, Ei and E* are conjugate, since they are both conjugate 
to E, and have the usual relations of object and image. 

4. The Chief Ray. The ray which passes through the system 
so as to intersect the axis at the plane of the aperture stop is 
called the chief ray. It is represented in Fig. 4-3 by the solid 
line OI f . The conjugate rays OA and I' A' will also intersect the 
axis at the planes of the two pupils, but will not necessarily 
intersect the axis at the centers of any of the lenses. The chief 
ray may be regarded as an axis of symmetry for the bundle of 
rays from a point which are restricted by an aperture. If the 



34 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. IV 



aperture is small, the chief ray may be used as a representative 
ray. 

6. Telecentric Systems. In case the aperture stop is placed 
at the principal focal point F of the lens, the chief ray after refrac- 
tion is parallel to the optical axis and the entrance pupil is at 
infinity. This system, illustrated in Fig. 4-4, is then said to be 



o ... 




FIG. 4-3. The chief ray of a bundle cuts the axis at the aperture stop. 

telecentric on the side of the image. Similarly, if the aperture stop 
is placed at the second focus F' of the system, it will be telecentric 
on the side of the object. The former system has certain advan- 
tages if the size of / is to be measured accurately, for the ^-position 
of /' will not depend upon its distance from the lens. Slight 
inaccuracy of focusing will result in blurring of the image point, 




Fio. 4-4. A system telecentric on the image side. 

but the center of the image will be the same distance from the 
axis as if it were accurately focused. This arrangement is of 
particular advantage in micrometer microscopes. 

Problems 

1. A thin lens of 3 cm. diameter and 6 cm. focal length is used as a 
magnifying glass. If the lens is held 5 cm. from a plane object, how 



APERTURES IN OPTICAL SYSTEMS 35 

far from the lens must the eye be placed if an area of the object 8 cm. 
in diameter is to be seen? 

2. A telescope has for its objective a thin positive lens of 20 cm. focal 
length and 5 cm. aperture, and for its ocular a thin positive lens of 

4 cm. focal length and 2 crn. aperture. Use a diagram and locate the 
position and size of the exit pupil, and the size of the field of view. 

3. A lens system whose entrance pupil is 25 mm. and exit pupil is 
20 mm. in diameter has a principal focal length of +12.5 cm. If an 
object whose height is 15 mm. is placed on the axis 30 cm. in front of 
the entrance pupil, where is the image, and what is its size? 

4. A camera has a thin lens whose aperture is 8 mm. and whose focal 
length is 10 cm. What is the //number of the system if a stop 7 mm. 
in diameter is mounted 5 mm. in front of the lens? If it is mounted 

5 mm. behind the lens? (The //number, or relative aperture, is the 
ratio of the focal length to the entrance pupil of the system.) 

6. Two thin lenses are placed 3.5 cm. apart. The first, nearer the 
object, has a focal length of +25 cm. and an aperture of 3.5 cm. diame- 
ter; the second has a focal length of 30 cm. arid an aperture of 4 cm. 
diameter. Which is the aperture stop for an object position 15 cm. 
from the first lens? If a stop with a diameter of 2.5 cm. is placed between 
them 2 cm. from the first lens, find the location of the aperture stop, 
the locations and apertures of the extranco and exit pupils for the object 
position given. What is the //number of the system? 

6. Using a diagram, describe a system which is telecentric on the 
side of the object. 



CHAPTER V 
PHOTOMETRY THE MEASUREMENT OF LIGHT 

1. Photometric Standards. The unit of luminous intensity 
of a source of light is the candle. If the candle power of a source 
is said to be 10, its luminous intensity is 10 candles. The 
standard candle was originally of sperm wax, weighing ^ lb., 
% in. diameter, and burning 120 grains per hr. The primary 
standards used in Great Britain, France, and the United States 
are specially made carbon filament lamps, operated at 4 watts 
per candle. In Germany and some other European countries 
the legal standard is the Hefner lamp, which burns amyl acetate 
and has an intensity of 0.9 U. S. standard candles when the flame 
is at a height of 40 mm. The unit of measurement of the light 
flux or flow of radiant energy from a source is the lumen. This is 
an arbitrary unit by which the flux is evaluated by its visual 
effect, and has the dimensions of power. The quantity of light 
radiated in any given direction from a point source of unit 
candle power into unit solid angle is 1 lumen. Hence the total 
luminous flux from a point source having unit candle power in all 
directions is 4w lumens. 

A source rarely radiates with the same flux in all directions. 
If the actual candle power is /, then the total luminous flux is 
given by 

/4ir 
/ rfw. (5-1) 



Hence we can define the luminous intensity, measured in candles, 

by 



/ = (5-2) 



If the mean candle power is /, F 4irl. 

At a distance r from the source let the light fall on a surface 
of area da, which subtends the solid angle rfw at the source, and 

36 



SEC. 5-2] PHOTOMETRY THE MEASUREMENT OF LIGHT 37 

whose normal makes an angle with the direction of the light as 
shown in Fig. 5-1 ; then, since the areS, da is given by 



J /IT 0\ 

da = --- -, (5-3) 

cos v ' 

it follows by comparison with eq. 5-2 that 

jr. T i Ida cos 6 ,_ . x 

dF = Jdw = ---- 3 --- (5-4) 

The illumination J? on a surface is defined as the flux per unit 

area; i.e., 

dF I cos e 

&=-== - ~ (o-o) 

da r 2 v ' 

In the metric system the unit of illumination is the lumen per 
square meter. 

du> _. ___ 




Fi. 5-1. 

A simple method for comparing the luminous intensities 
(candle powers) of two point sources is at once evident. If two 
sources I\ and /2, at distances ri and r 2 respectively from a screen 
on which the light is incident at the same angle 6, produce on the 
screen equal illumination, then 

fl = ?? (fM}) 

The experimental determination of equality of illumination 
either by the eye or by some auxiliary device is a matter of con- 
siderable difficulty. This is especially true when the illumination 
is either very faint or very strong, or when the sources do not 
have the same color. The measurement of relative illumination 
is called photometry. If the measurement takes into account the 
wave-length of the light it is called spectrophotometry.f 

2. Brightness of Extended Sources. If the source is not a 
point, but is of appreciable size, it is customary to speak of its 
brightness instead of its intensity. Brightness is denned as the 
intensity per unit area of the source, measured in candles per 



38 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. V 

square centimeter in metric units. If B is the brightness, the 
intensity in a direction making an angle a with the normal to 
the radiating surface is given by 

7 = B ds cos a. (5-7) 

Substituting in eq. 5-7 the value of 7 given by eq. 5-4, it follows 
that the flux through the solid angle subtended by the area da in 
Fig. 5-2 receiving the light is 

B ds da cos a cos B 






, 



,_ _ x 
(5-8) 



where is the angle between the normal to da and the direction 
of the light. Some luminous surfaces do not radiate uniformly 
in all directions, so that rigorously the variation of B with a 
should be taken into account. In what follows it is assumed that 
B is independent of a. 





FIG. 5-2. 

The term "brightness" is also used to mean the intensity of 
reflection of a diffusely reflecting surface. Such a surface has the 
same brightness at every angle of observation. Similarly, a 
radiating surface which has the same brightness in every direction 
is called a diffusely radiating surface. 

Brightness may be measured in lamberls as well as in candles 
per square centimeter. The brightness of a perfectly diffusing 
surface which radiates or reflects 1 lumen per sq. cm. is 1 lambert. 

3. Lambert's Cosine Law. Consider a radiating sphere for 
which every element of surface has the same brightness. As seen 
from a point F, Fig. 5-3, whose distance r away from the sphere 
is large compared to the diameter of the sphere, it will appear as a 
flat disk. The flux from an element of area ds, at the center of 
this disk, falling normally upon an area da at P, will, by eq. 5-8, be 

B dsi da 



Also, the flux which appears to come from another element of the 
same size on the disk will in reality be that from an element ds 2 
of the sphere, and will be 



SBC. 6-4] PHOTOMETRY THE MEASUREMENT OF LIGHT 39 

B ds 2 da cos a 



But since ds 2 = rfsi/cos a, e^ 2 = dFi. If rfa is the pupil of the 
eye, it follows that a sphere radiating with the same intensity 
over its entire surface will appear as a disk of uniform brightness. 
It also follows that an element of surface ds on the sphere will 
have the same apparent brightness when observed from any 
direction, provided the point of observation is the same distance 
away. A luminous surface with these properties radiates accord- 
ing to Lambert's cosine law, which states that the intensity from a 
surface element of a diffuse radiator is proportional to the cosine 
of the angle between the direction of emission and the normal 
to the surface. This law may also be applied to diffusely 
reflecting surfaces. 



Apparent disk 




P 

Distant 
point 



FIG. 5-3. Illustrating Lamberts' cosine law. 

4. Photometric Principles Applied to Optical Systems. In 

optics it is sometimes necessary to know the illumination of an 
image formed by an optical system. A knowledge of the entrance 
and exit pupils is important. Suppose we wish to find the total 
light from a surface ds of brightness B through a system whose 
entrance pupil radius subtends an angle U, as in Fig. 5-4. Con- 
sider at the entrance pupil a ring cut by two cones whose apices 
are at ds and whose generating lines make angles a and a -f- da 
with the normal to ds. If the distance r is unity, the area of this 
ring is 2ir sin a da. 1 The solid angle subtended by this ring is 
da, which, by eq. 5-3 is equal to (da cos 0)/r 2 . Substituting in 
eq. 5-8 for this quantity its equivalent, 2ir sin a da, it follows that 
the radiation through the ring is 

1 The area of a ring of width w whose mean radius is a is 2iraw. In the 
case illustrated in the text, a = sin a and w = da. This result neglects a 
second order term proportional to (da) 8 . 



40 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. V 



dF = 2irB ds da cos a sin a (5-11) 

and the total luminous flux through the pupil is 

f U 

Fu = 2irB ds I sin a cos a da = irB ds sin 2 C7. (5-12) 
jo 

Similarly, if we consider the image ds' of ds to be formed by a 
system whose exit pupil has a radius subtending an angle V, 
corresponding to an entrance pupil of radius U, the luminous 
flux through the exit pupil is 

FV - irB' ds' sin 2 U', (5-13) 

where B' is the brightness of the image ds'. Assuming the 



da 




. 5-4. 



transmitting media to be transparent, 

, * 

B ds sin 2 U = B' ds' sin 2 U 1 . (5-14) 

Because of light absorption and reflection from the lens surfaces, 
in actual practice the right-hand member of the equation will be 
the smaller of the two. 

It can be proved that in a so-called aplanatic system, for a 
single position on the axis 

ny sin u = n'y' sin u' = constant (5-15) 

for any number of media in an optical system. The angles 
u and u' are those made with the axis by a pair of conjugate rays; 
hence they can be identified with U and U' t the rim rays to the 
boundaries of the entrance and exit pupils. In eq. 5-14, since 
ds and ds' are elements of area, we may write 



and consequently, 



* _ (y.\ 
*t ~ v/ 



SBC. 5-6] PHOTOMETRY THE MEASUREMENT OF LIGHT 41 

B' w' 2 

~B = ~tf' (5 ~ 16 > 

Equation 5-16 says that if it were possible to construct a lens 
system in which there are no losses by absorption and reflection, 
the brightness of the image is at best equal to that of the object, 
provided also that n' n. 

This may be further exemplified as follows : ^uppose an area 
A in Fig. 5-5 to be illuminated by a source <S. Its brightness will 
be given by eq. 5-4 or 5-8. The interposition of a condensing 
system at B will increase the intensity of illumination at A by 
concentrating the light intercepted by B on a smaller area, 
provided the losses by absorption and reflection at B are not too 
great. But the same increase could be obtained by bringing the 
source nearer to A. We can draw the important conclusion that 
no device for concentrating the light from a source can produce an 




Fio. 5-5. 

intensity of illumination in the image which is as great as that which 
would result from putting the same source at the image position. 

6. Numerical Aperture. From eq. 5-15 it follows that the 
quantity of light entering the instrument depends on rc 2 sin 2 U. 
Abbe called the quantity n sin U the numerical aperture (N.A.) of 
a system. In telescopes and cameras, another quantity called 
the relative aperture 1 is given by the ratio of the focal length of the 
system to the diameter of the entrance pupil. The choice of this 
designation depends upon the fact that in such instruments the 
object is either at a great distance or at infinity. 

6. Natural Brightness. It is important that we distinguish 
between the amount of light which falls on a screen from a 
luminous source and the brightness of the source as seen by the 
eye. The former, which has been discussed in Sec. 5-4, is the 
brightness of the surface on which the light falls. If the source 
is observed with or without the aid of other optical systems, the 
image is formed on the retina of the eye. In case the unaided eye 
is used, it follows from eqs. 5-13 and 5-16 that the quantity of light 
falling on unit surface of the retina is 

1 Or // number. 



42 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. V 

E - TrBn 2 sin 2 F, (5-17) 

where n is the index of refraction of the vitreous humor and V 
is the angle subtended by the radius of the exit pupil at the retina, 
which is in this case the pupil of the eye. The quantity Bn 2 in 
eq. 5-17 is substituted for B f in eq. 5-13. 

By eq. 5-17 we see that the brightness E of the object as seen 
by the eye is independent of the distance of the object, but 
depends on B and V. It is called the natural brightness. If 
the pupil of the eye enlarges, the natural brightness is increased. 

7. Normal Magnification. If the light is received by the 
eye with the aid of an external optical system, we can regard 
the whole as a single system for which the foregoing will be true. 
There will, however, be two cases, depending on the relative 
sizes of the exit pupils of the external system and of the eye : (a) 
When the exit pupil of the external system, whose radius sub- 
tends an angle which may be called V, is larger than the exit 
pupil of the eye, then the limitation on the natural brightness is 
imposed by the pupil of the eye and equation 5-17 holds. (6) 
When the exit pupil of the external system is smaller than the 
exit pupil of the eye, V will limit the brightness on the retina, 
which will be given by 



E' = T#n 2 sin 2 V. (5-18) 

Hence, from eqs. 5-17 and 5-18, for small angles 

E' F'* 



If the object has an extended area, so that the angle it subtends 
at the unaided eye is greater than F', the brightness will be no 
greater than that of an object with exit pupil whose radius 
subtends an angle F. Hence for sources of large area the external 
optical instrument does not increase the brightness of the image 
but merely increases the visual angle. 

In the case of a microscope, where the radius of the exit pupil 
is smaller than the radius of the pupil of the eye, the numerical 
aperture is of great importance. For small angles, the radius 
of the exit pupil may be represented by d sin U r , where V is the 
angle subtended by the radius of the exit pupil of the optical 
instrument and d is the distance from the object to the pupil of 



SEC. 6-8] PHOTOMETRY THE MEASUREMENT OF LIGHT 43 

the eye, the latter being placed at the exit pupil of the instrument. 
If h is used for the radius of the pupil of the eye, then 

E' _ d* sin* V 

~E -- T? -- (5 ' 20) 

Here E is the brightness without, and E' is that with, an external 
instrument. By eq. 5-15, it follows that 



sin 2 V = . sin 2 V. 
n' y' 

Since y' /y 2 = |3 2 , where /3 is the lateral magnification, eq. 5-20 
may be written 

E' _ d V sin* U 
~E - 



whence, since n 2 sin 2 (/= [N.A.] 2 , it follows that E'/E is pro- 
portional to the square of the numerical aperature. Hence for 
the greatest brightness E' it is necessary to have as large a 
numerical aperture as possible. Also, for a numerical aperture 
of a certain size it is possible to have a magnification such that 
E' equals the natural brightness E. A magnification of this 
amount is called the normal magnification. 

8. Effects of Background. For point sources and those of 
very small area, the foregoing rules do not hold, principally 
because of departures from the laws of rectilinear propagation. 
When small angular apertures such as that of the eye are used, 
diffraction plays an important part. Roughly speaking, the size 
of the image of a point source on the retina depends inversely on 
the size of the pupil of the eye. When a star is seen with the 
unaided eye, the light enters an area irh z ', if with a telescope, the 
light enters an area ira*, where a is the radius of the telescope 
objective. If the exit pupil of the telescope is less than or equal 
to the pupil of the eye, all the light passing through the objective 
enters the eye. Hence the effect on the retina will be an increase 
on the brightness of the star in the ratio a 2 / A 2 , where A is the 
radius of the exit pupil of the telescope. If the exit pupil of the 
telescope is greater than h, not all of the light enters the eye, and 
in this case the increased brightness will be a 2 /h 2 times that with 
the unaided eye. In either case, there will be an increase in the 



44 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. V 

brightness of the starlight. On the other hand, as shown in 
Sec. 6-7, the brightness of the background of the sky is not 
increased. If the magnification is greater than the normal 
magnification, it may even be diminished. Thus, with a telescope 
we can see stars of smaller luminosity than with the naked eye, 
even when a considerable amount of skylight is present. If 
nearby objects are viewed, the length of the telescope must be 
small in comparison to the distance of the object for an increase 
of brightness to be obtained. 

In dealing with vision, the physiological aspects should not be 
neglected. Especially in the case of persons who are color-blind 
or partly so, objects of different sizes differ in their visibility. 
The effects of irradiation must also be taken into account. From 
the bottom of a deep shaft stars can be seen even in broad day- 
light. Here the starlight has not been reduced, neither has the 
brightness of the sky, but merely the total light sent into the 
eye from the whole sky. Objects not distinguishable in a dim 
light may be seen more easily by restricting the vision to those 
objects by masking off the light of nearby brighter areas. Irradi- 
ation of the eye by ultraviolet light from an otherwise invisible 
source will also serve to obscure the vision of surrounding objects. 

The aperture of an optical instrument also serves to limit the 
ability to see separately objects which are close together, i.e., it 
determines the resolving power of the instrument. The sub- 
ject of resolving power will be discussed in the chapter on 
diffraction. 

Problems 

1. Find two points on the straight line joining two sources where the 
illumination is the same. The sources are 20 candle power and 30 candle 
power, respectively, and are 300 cm. apart. 

2. A simple lens having a diameter of 8 cm. and a focal length of 
25 cm. is used to focus the light of the sun on a white screen. What is 
the ratio of the brightness of the image to the brightness when the screen 
is illuminated by the sunlight without the use of a lens? 

3. A lamp whose intensity is 75 candles is placed 300 cm. from a 
screen whose reflecting power is 70 per cent. If the screen is a diffuse 
reflector, what is its brightness in candles per cm. 2 ? In lamberts? 

4. Why does a celestial telescope enable us to see stars brighter by 
contrast with the background of the sky? 



CHAPTER VI 

ABERRATIONS IN OPTICAL SYSTEMS 

There are five aberrations, or faults, in ordinary lens or mirror 
systems, which are due to the shapes of the surfaces employed, 
the relative positions of the stops, or the position of the object: 
spherical aberration, astigmatism, coma, curvature of the irnag^ 
field, and distortion of the image. To these may be added, for 
lenses but not mirrors, the fault called chromatic aberration, 
which is due to the variation of index of refraction of transparent 
substances with color. Spherical and chromatic aberration occur 
even in the case of point objects on the axis of a lens system, 
while the other four astigmatism, coma, curvature of the field, 
and distortion occur in the case of point objects off the axis. 

If the angle made by any ray with the axis is u, the assumption 
has been made in the theory of ideal optical systems that 
sin u = u. This assumption leads to the so-called first-order 
theory. The expansion of sin u into a series results in 



U 1 



. //. ,\ 

sm u = u g-j -f ^ ^TJ + (6-1) 

The extent of the departure from ideal theory depends upon the 
extent to which terms in odd orders of u must be added; this in 
turn depends either upon the size of the aperture of the lens, or 
the distance of the object point from the axis, or both. The 
rigorous mathematical analysis of these aberrations to the third 
and higher orders has been made the subject of a great deal of 
study. Indeed, the subject is one still engaging the attention of 
specialists in the field of optics, and a great deal of progress is 
being made in the development of new methods for reducing 
these aberrations to a minimum in optical systems. Although 
the subject is one which is too extensive to be mastered by any 
but highly trained specialists, the fundamental ideas involved 

are relatively simple. 

46 



46 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VI 

The most comprehensive analysis of the five aberrations was 
made by von Seidel, who developed a group of five terms for their 
correction. 1 These terms are to be applied to first-order theory 
for ideal optical systems to take into account the third-order 
corrections for rays making appreciable angles with the axis. 
When the oblique rays fulfill the same conditions as the paraxial 
rays, the terms become zero. However, the equations thus 
obtained cannot be solved explicitly for the radii of curvature of 
the refracting surfaces, so that in practice it is more expedient 
to trace the path of each ray through the optical system and in 
this manner find the appropriate surface curvatures for the reduc- 
tion of the aberrations to the required degree. 
( 1. Spherical Aberration. The equations for refraction in ideal 
optical systems given in Chap. II were derived on the assumption 




FIG. 6-1. Illustrating spherical aberration. 

that the aperture of the refracting surface was sufficiently small 
so that distances from the object to points on the surface could 
be considered equal. This was also assumed with regard to 
distances from the image to the refracting surface. For any 
optical system, the departure from this equality will depend on 
the size of the aperture used. On refraction at a spherical sur- 
face, as the ratio of aperture to focal length is increased, the 
rim rays, i.e., those which are refracted at the boundary of the 
surface, will converge to an image point considerably closer to 
the surface than will the paraxial rays, which are those lying 
extremely close to the axis throughout their lengths. The 
point /' in Fig. 6-1, to which the paraxial rays converge, is called 
the Gaussian image point. Each rim ray extended beyond the 

1 A simple treatment of the von Seidel equations is given in Whittaker's 
"Theory of Optical Instruments." 



SBC. 6-1] ABERRATIONS IN OPTICAL SYSTEMS 



47 



axis cuts the caustic (the envelope of all rays of different slopes) 
at N, where the diameter of the circular cross section of the entire 
bundle of rays has its minimum value in the range from L to /'. 
This area is called the least circle of aberration. It can be reduced 
in size by diminishing the aperture, at the expense of illumination, 
by changing the shape of the surface, or by combining several 
refracting surfaces which mutually compensate for the aberration. 
The last two methods may introduce other defects in the image, 
so that in most cases a compromise must be effected which will 
yield the result most satisfactory for the purpose of the particular 
optical system. 




(a) (b) 

Fia. 6-2. Demonstration of the effect of spherical aberration in a single lens, 
(a) is a photograph of a screen, (b) is a photograph taken of a point source of 
monochromatic light with the lens covered by the screen. The photographic 
plate was placed at the Gaussian image point. Only the rays through the large 
central hole in (a) are in focus in (b). 

An excellent illustration of spherical aberration can be made 
with an ordinary plano-convex lens. Figure 6-2a is a photograph 
of an opaque screen having a hole in the middle and smaller 
holes in zones at different distances from the axial position. 
With this screen placed over the lens and a point of light about 
1 mm. in diameter as a source, the photograph in Fig. 6-26 was 
made. A filter was used to render the light nearly mono- 
chromatic. The image at the center corresponds to the Gaussian 
image point and is formed by the rays through the central hole. 
The rings of images about this point show the rapid increase of 
spherical aberration for zones of larger radius. If the screen were 



48 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VI 

not over the lens, the resulting image would be a bright image 
point at the center, with a circular area of smaller illumination 
about it, fading rapidly to its periphery. 

2. Third-order Corrections to Spherical Aberration. The 
third-order correction may be found algebraically without an 
undue amount of labor, but the calculation of terms of higher 
order is an extremely laborious process. Since the method used 
in getting the third-order term involves little of interest beyond 
the final result and the approximations involved, it is given in 
Appendix II, and the result alone is given here. The approxima- 
tions depend upon the appropriate simplification of intermediate 
equations. 1 

The introduction of the third-order correction for a thin lens 
results in the equation 



a a k 



,. (_,)(!-!)+ (6-2) 



n 1 htf ( 1 l\Yn + 1 1\ /I 1 
w*~~ ' ~2\ \7 a/ V a F/ \r a 7 




in which a' is the distance of the image from the center of the 
lens, and a*/ is the image distance for an oblique ray cutting the 
refracting surface at a distance h from the axis. From a com- 
parison of eq. 6-2 with the first-order equation 

- + -^ = (n - l)(- - -} (8-2o) 

a a' '\ri r 2 / 

it is evident that the term for the lateral spherical aberration is 

n - 1 A* //I 1\Y + 1 1\ /I lV/n + 1 l 
"n ' 2 )V, + a) \~^T + 7j + \Tt ~ J) \~J~ ~ 7 

(6-3) 

for rays incident upon the lens at a distance h from the axis. 
Since the quantity given in 6-3 varies as h 2 , it increases rapidly 
with an increase in the aperture of the lens. 

The longitudinal spherical aberration in a thin lens, the radius 
of whose aperture is h t is given by a*' a'. This may be 
obtained by subtracting eq. 6-2 from eq. 6-2a, which gives 

1 A very complete discussion of the algebraic corrections to the third and 
higher orders is found in H. Dennis Taylor, "A System of Applied Optics." 



SBC. 6-3] ABERRATIONS IN OPTICAL SYSTEMS 49 



where [S.A.] is written for the lateral spherical aberration, or, 



a k ' - a' = -a 

If the difference between a' and a*/ is not too great, the last 
equation may be written 

a k ' - a 1 = -' 2 [S.A.]. 

This is the difference between the focal lengths of the rim rays 
and paraxial rays. In using these equations, it should be remem- 
bered that by convention the radius r 2 of the second surface of a 
double-convex lens is negative. 

3. Coddington's Shape and Position Factors. Coddington 
has obtained an expression for the spherical aberration of a thin 
lens in terms of two quantities which we may denote by s and p, 
factors representing respectively the "shape" of the lens and the 
position of the object. The values of these factors in terms of 
known constants are stated as follows: In the first-order equation 
for a simple lens, eq. 6-2a, let 

1 _ (1 + p) I _ (1 -p) 

a " ~2f ' a' 2f " 



"' 



n 2/(n - 1) r, 2f(n- 1) 

Substituting these in 6-3, the lateral aberration becomes 

(3n 



Differentiating with respect to s, we obtain 

d[S.A.] _ h* \2(n + 2)8 + 4(n - l)(n + I)p1 
"~5T" " V 5 ' I ^^T) r J' 

which becomes zero when 



(6-6) 



50 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VI 

Thus, for a lens to have a minimum lateral aberration for 
distant objects (p = 1), eq. 6-6 imposes on the surface curva- 
tures the condition 

r\ _ s 1 _ 2w* n 4 
r^ ~ 7+~I n + 2w 2 ' 

If the index of the glass is 1.52, s = 0.744, and n/r 2 = -0.148; 
i.e., for the most favorable form of convex lens the radius of 
curvature of the surface toward the object is about one-seventh 
of that toward the image. For a plano-convex lens with the 
curved surface toward the object the spherical aberration is almost 
as small as for a lens whose radii of curvature have the ratio 
given in eq. 6-7, but when the plane side of the lens is toward 
the object, the aberration is very large. 

In the foregoing, it is assumed that the lens is so thin that its 
thickness has no appreciable effect. For thick lenses, special 
allowance must be made in correcting for the spherical aberra- 
tion. It can never be eliminated entirely for a single lens, but a 
combination of a convergent and a divergent lens can be made 
for which the aberration is zero. 

Since the index of refraction varies slightly with the wave- 
length, it is evident that there is some dependence of spherical 
aberration on the latter. It is not usual to take this into account, 
however, since in most cases the effect is small compared to the 
ordinary aberration. 

4. Astigmatism. When light spreads out from a point source, 
the wave front is spherical in form if the medium is isotropic and 
homogeneous. The wave front retains its symmetry if inter- 
rupted by a refracting or reflecting surface, and, if the point 
object is on the optic axis, the rays will converge to a point 
image provided spherical aberration is absent. If we consider 
only the rays refracted or reflected by a narrow ring-shaped zone, 
with its center at the vertex, the cross sections of the beam at 
various axial positions will be as shown in Fig. 6-3. 

If, instead, the point source is not on the axis, the alteration of 
the curvature of the wave front upon refraction or reflection will 
not be symmetrical even in the absence of spherical aberration, 
and the rays will not converge to a single point image. This 
lack of symmetry will also exist for a point object on the axis if 
the surface is not symmetrical with respect to the axis, i.e., if 



SEC. 6-5] ABERRATIONS IN OPTICAL SYSTEMS 



51 



it is not a surface of revolution about the axis. In either case, 
the resulting image will be astigmatic, and the cross sections of 
the wave front at positions near the focus will be as shown in 
Fig. 6-4. When astigmatism is present, there are two line foci 
at right angles to each other, while the closest approach to a 
point image is a circular patch or confusion of light between them. 
Sometimes a distinction is made between the astigmatism pro- 
duced by oblique rays, as described above, and that produced by 




OOo o _ 
image point 

FIG. 6-3. Showing the shapes of stigmatic bundles before and behind the image 

point. 

the refraction or reflection by cylindrical surfaces. In the latter 
case, there is merely one focal position, so that the image of a 
point source is drawn out into a line parallel to the axis of the 
cylinder. For simplicity, only the first case will be discussed, 
as it is more definitely classified as an aberration. 

5. Primary and Secondary Foci. The two line foci shown in 
Fig. 6-4 are known as the primary and secondary foci, the former 




0>o> - o O 



Fio. 6-4. Showing the shapes of astigmatic bundles before and behind the two 

astigmatic line images of a point object. 

being nearer to the system in the illustration. The primary 
focus is sometimes called the meridional and sometimes the 
tangential focus, while the secondary is sometimes called the 
sagittal focus. 

The equations giving the distances from a single refracting 
surface to the two astigmatic image positions are derived in 
Appendix III. They are, for the primary and secondary foci, 
respectively, 



52 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VI 



n cos 2 i , n' cos 2 i' n' cos i' n cos i 



s 



Si - r 

?i n' _ -n 7 cos i' n cos i 
8 so' r 



(6-8) 



in which i and i 7 are the angles of incidence and refraction, 
respectively, n and n' are the indices of refraction of the first 
and second media, r is the radius of curvature of the surface, is 
the distance from the point source to the surface, and s\ and 8% 
are the distance from the surface to the primary and secondary 
astigmatic images, respectively. For a spherical mirror, these 
equations reduce to 



(6-9) 



Coddington has shown that for a thin lens in air with a small 
aperture stop the following equations give the positions of the 
astigmatic foci. The conventions regarding the signs of r\ 
and TZ are the same as those previously used. 



1 


1 


2 


1 




Si 


r cos i 


/ cos i' 


1 


1 


2 cos i 


_ cos i 


s 


S'o' 


r 





l + i = .JJl _ lY."-^ _ A 

s i cos i\r\ t'z/ \ cos t / 

1,1 /I iVncosi' A 
= cos il II . II- 

&2 \?"i r2/\ COS ? / 



(6-10) 



These equations reduce at once to the ordinary formula for a 
thin lens in air if i and i' become zero whereupon the astigmatism 
disappears. 

6. Astigmatic Difference. The difference between the dis- 
tances from the lens of I\ and 1% is called the astigmatic difference; 
it is found by subtracting the value of i 7 from that of s 2 7 . For 
the mirror the astigmatic difference is 



Si 7 = 2si 7 s 2 7 sin i tan i, 



(6-11) 



from which it can be seen that the difference increases rapidly 
with the angle of incidence. This is also true for lenses. Since 
this defect is due to the angle of incidence of oblique rays upon 
the surface, it is evident that its form will be different for diver- 
gent systems. In Fig. 6-5 are shown characteristic positions 
of the loci of orimarv and secondary foci for convergent and 



SBC. 6-7] 



ABERRATIONS IN OPTICAL SYSTEMS 



53 



divergent systems. l From this figure it is evident that combina- 
tions of systems may be made in which the astigmatic differences 
compensate for one another to some extent. In the photographic 
anastigmat combination, not only is the astigmatism but also 
the curvature of the field largely eliminated over a considerable 

P S 



o 
>*> 



JQ 
O 



PC S c 




25- 
f 20- 

*X 

cs 
15- 

i 

en ,^o 




Distance from lens - 



9.9 10.0 10.1 
Distance from lens in cm.- 
FIG. 6-5. FIG. 6-6. 

FIG. 6-5. Loci of astigmatic focal positions for convergent and divergent 
lenses. 

FIG. 6-6. Showing the loci of positions of astigmatic images at different 
angles with axis for a corrected photographic lens of 10 cm. focal length. 

area in the image plane. A diagram of the focal positions for 
this combination is shown in Fig. 6-6. 

7. Coma. A system is corrected for spherical aberration 
when rays from an object point all intersect at the same point. 
This may be effected for axial points, while for objects having 
appreciable area there may still be a variation of lateral magnifi- 
cation with zonal height h as illustrated in Fig. 6-7. Moreover 
the rays contributing to the image which pass through the lens 
at a distance h from the axis may pass through the focal plane, 
not at a common point, but in a circle of points, the size of the 
circle depending on the radius of the zone and several factors in 
the construction of the system. Figure 6-8 illustrates the forma- 
tion of the so-called comatic circles. The numbers on the largest 
circle correspond to numbered pairs of points on a zone of the 
lens, indicating the origin of the pair of rays which intersect at 
each point on the comatic circle. The heavy line PI represents 

1 The shapes of the focal curves vary also with stop positions, and not 
necessarily with focal length. 



54 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VI 



the chief ray of the bundle. Each zone of the lens produces a 
comatic circle, the radius increasing as h increases. The centers 
of the comatic circles will also be displaced, either toward or 




FIG. 6-7. Illustrating pure coma. 

away from the axis. In the case illustrated, the resulting flare of 
the pear-shaped image is away from the axis, and the coma is said 
to be positive. If the flare is nearer the axis than the image point 
of the chief ray, the coma is negative. 




Fio. 6-8. The formation of comatic images. Pairs of rays from a given zone, 
such as 1 and 1, 2 and 2, etc., meet at points not common to all rays, but lying on 
a comatic circle whose distance from the axis varies with the radius of the zone. 

Since the condition which results in coma is a difference of 
lateral magnification for rays passing through different zones of a 
lens, the constancy of y'/y for all zones will result in its elimina- 
tion. It can be shown that provided y' and y are small distances 
in the object and image planes, 



SBC. 6-9] ABERRATIONS IN OPTICAL SYSTEMS 55 

n'y' sin u' = ny sin u, (6-12) 

where u and u' are the angles between conjugate rays and the 
axis. 1 Hence the magnification will be constant and coma will be 
absent, provided sin u'/sin. u is constant. This is known as 
Abbe's sine condition. For small angles u and u', it is the same 
as LaGrange's law. 

Figure 6-9 is a photograph of a region of the sky taken with a 
24-in. reflecting telescope. The effect of coma shows in stellar 
images which are some distance from the center of the field. 

For a very distant object near the axis coma will be absent if 

r 

- -. = constant. (6-13) 

sin u ^ ' 

This equation is easily derived from eq. 6-12. 

8. Elimination of Coma. It can be shown that the condition 
for no coma, i.e., the sine condition, can also be stated in the 
terminology of Coddingtoii as 

s(2n + l)(n - 1) + (n + l)p = 0, (6-14) 

in which s and p are, respectively, the shape factor and the position 
factor as before. Since this equation is linear in s, it is possible 
to eliminate coma entirely from a lens system for a single object 
position. A lens system which is corrected for both spherical 
aberration and coma for a single object position is called aplanatic. 
It can be shown 2 that the condition for no spherical aberration 
for two positions P\ arid P- of the object, when they are near 
each other on the axis, is 

sin 2 



where Pz and Pi are the images of P 2 and PI. Since this con- 
dition and the sine law cannot be true at the same time, an optical 
system cannot be made aplanatic for more than one position of 
the object. 

9. Aplanatic Points. Two points on the axis which have the 
property that rays proceeding from one of them shall all con- 
verge to, or appear to diverge from, the other are called aplanatic 

1 For a simple proof of the sine law, see Drude, "Theory of Optics," pp. 
58 and 505, in the English translation. 

2 See Drude, "Theory of Optics," p. 62 of the English translation. 



56 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VI 



o 




SEC. 6-9] 



ABERRATIONS IN OPTICAL SYSTEMS 



57 



points. A useful device for describing their properties, originally 
discovered by Thomas Young, although later independently 
discussed by Weierstrass, is illustrated in Fig. 6-10. Light from 
a medium of index n is refracted at a spherical surface into a 
medium of index n'. The surface is given by a circle drawn 




FIG. 6-10. Young's construction for refraction at a spherical surface. 

concentrically with two circles whose radii are equal to rn/n' 
and rn'/n, where r is the radius of curvature of the surface. The 
projection of an incident ray cuts the larger circle at point M , and 
a line drawn from M to the center C cuts the smaller circle at AT. 




FIG. 6-11. Axial aplanatic points in refraction. 

A straight line from the point of incidence A through AT is the 
refracted ray. The construction of a few such rays, incident on 
the surface at different distances from the axis, will readily illus- 
trate that they cannot intersect in a single given point. If how- 
ever, the points M and N are on the axis, as illustrated in 



58 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VI 



Fig. 6-11, the refracted rays will all meet at N. Conversely, if 
rays originate at N, after refraction they will appear to come from 
a virtual source M. In this case the points M and N are called 
aplanatic points of the refracting surface. They have the prop- 
erty that rays originating at one of them will be refracted so as to 
pass through or be projected back through the other. These 
points have an important practical application in the construc- 
tion of microscope objectives. As illustrated in Fig. 6-12, the 
lens closest to the object is made hemispherical, with the flat 
surface near the object. Light from P, the object position, will 
be refracted so that there will be a virtual image at P'. The 

medium between P and the spheri- 
cal surface is made practically con- 
tinuous by immersing the object in 
an oil of index of refraction about 
the same as that of the glass. 

The lateral magnification of the 
image at P' will be P'V/PV. If a 
second lens Lz is added in the form 
of a meniscus with its 




concave 



FIG. 6-12. The principle of 

aplanatic points applied to a micro- sp herical surf ace having a center of 

scope objective. r , .,, , 

curvature at P , there will be a 

second refraction at the rear, convex, surface, giving rise to a vir- 
tual image at P" with a second lateral magnification. There is a 
limit to which the magnification can be repeated in this manner, 
because of the introduction of chromatic aberration. 
An aplanatic refracting surface has the equation 

na n'a' = constant, 

in which a and a' are the distances from the object and image, 
respectively, to the surface. This is the equation of a Cartesian 
oval. For an aplanatic reflecting surface, the equation is 
a a ' = constant. This is the equation of an ellipsoid of 
revolution about the line joining the object and image points. 

10. Curvature of Field. It has been shown that for an object 
point not on the axis there are two line or astigmatic foci. If 
the object is an extended plane, the astigmatic images will not 
be planes, but curved surfaces. For object points on or near 
the axis, there will be sharp point-to-point representation in the 
image plane, but as the distance from the axis is increased, the 



SBC. 6-10] ABERRATIONS IN OPTICAL SYSTEMS 59 

sharpness of the image will decrease. Instead, each point of 
the object will be represented by a blurred patch, the size of which 
will be greater for greater distances from the axis. Even if the 
defects of spherical aberration, astigmatism, and coma are cor- 
rected, this patch will be a circle of confusion and will be the 
closest approach possible to a sharp-point focus. The surface 
containing this best possible focus for all parts of the image will 
not be a plane, but a surface of revolution of a curved line about 
the axis. This defect is known as curvature of the field. The 
condition for its removal was first stated by Petzval. While this 
condition may be applied to systems composed of a number of 
lenses, for a pair of thin lenses in air it reduces to 

=0- (6-15) 



For a convergent combination in which /i is the focal length 
of the positive, and fz is the focal length of the negative com- 
ponent, / 2 must be greater than f\. Therefore, in order that 
eq. 6-15 may be satisfied, it is necessary that n 2 be less than n\. 
In the earlier days of the past half century it was not possible to 
fulfill this condition for an ordinary achromatic doublet. Such 
a doublet is made of a convergent lens of crown glass in contact 
with a divergent lens of flint glass, the reason for this combination 
being that the flint glass, having a higher index of refraction, also 
has higher dispersive power necessary for the correction of 
chromatic aberration. About 50 years ago, however, under the 
leadership of Abbe, there were developed at the Jena glass works 
certain kinds of glasses for which, in a given pair, the one having 
a higher index had the lower dispersive power. With these 
glasses achromatic doublets can be made which also have a flat 
field free from astigmatism. 

Astigmatism may be corrected to a considerable extent by the 
use of an aperture stop which will limit each bundle of rays to 
those in the neighborhood of the chief ray from any object point. 
Similarly, curvature of the field may also be corrected. The 
proper use of a front stop is made in certain kinds of inex- 
pensive cameras to reduce curvature of the field, at the expense 
of aperture. Usually a meniscus lens is employed, as illustrated 
in Fig. 6-13. While for objects off the axis there is some astig- 
matism, by the proper location of the aperture stop it is possible 



60 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VI 



to obtain fairly good images on a flat field. This is due to the 
fact that at all parts of the field the circles of least confusion, 
midway between the astigmatic image surfaces, lie very nearly 
in a plane. 

To eliminate both curvature of the field and astigmatism or, 
rather, to correct them to a suitable degree simultaneously, it is 
necessary to use at least two thin lenses. In photographic 




Fio. 6-13. Astigmatic primary and secondary focal planes for a meniscus lens. 

objectives, where the elimination of these defects is desirable, the 
lens combination is sometimes a triplet of two convergent lenses 
and one divergent lens. 

11. Distortion. One of the requirements of an ideal optical 
system is that the magnification is to be constant, no matter at 
what angle the rays cross the axis. The failure of actual systems 
to conform to this condition is called distortion. The introduction 




Fio. 6-14. The pinhole optical system. 

of a stop, useful in reducing astigmatism and curvature, will also 
aid in correcting distortion. If an image is formed by means of a 
pinhole in a screen, the magnification will be constant, as shown 
in Fig. 6-14, since each pair of conjugate points in the object and 
image planes will be joined by a straight line. This constancy 
of magnification can be expressed by the equation 

tan u f 



tan u 



= constant 



SEC. 6-12] ABERRATIONS IN OPTICAL SYSTEMS 



61 



for all values of u. If a lens is used in place of the pinhole, there 
will still be constant magnification, or, as it is called, rectilinear 
projection, provided the lens is sufficiently thin. For an ordinary 
lens, the presence and location of a stop will make a considerable 
difference in the amount and character of the distortion. 

If the lens system is made of two symmetrically placed elements 
with the aperture stop midway beween them, the entrance and 
exit pupils will be at the principal planes of the combination. 
This system is free from distortion for unit magnification. For 
other magnifications, on account of the large angles of incidence 
for points far from the axis, spherical aberration will be present. 
The emergent ray, traced backward, will seem to come from a 
point Pi' not coincident with the emergent principal plane. 
Similarly, for large angles, the incident ray will intersect the axis 




Fu. 6-15.-- A symmetrical doublet. 
\ 

at Pi, near P. Only for paraxial rays will the chief ray of any 
bundle intersect the axis at the principal planes, as shown in 
Fig. 6-15. The result is that the system of rays from an extended 
object will suffer a change of magnification with increasing dis- 
tance from the axis. To be free from the resulting distortion, the 
system must be corrected for spherical aberration with respect 
to the pupils and must fulfill the condition that tan u'/t&n. u = 
constant. Any system thus corrected for both distortion and 
spherical aberration is called an orthoscopic or rectilinear system. 

Since the change of magnification present in distortion may be 
either an increase or a decrease, there are two kinds of distortion, 
illustrated by diagrams in Fig. 6-16, and by photographs in 
Fig. 6-17. 

12. Chromatic Aberration. In the development of simple 
lens theory, the variation of index of refraction with wave- 
length was ignored. While this variation can be turned to useful 



62 



LIGHT: PRINCIPLES AND EXPERIMENTS [HAP. VI 



account in prismatic dispersion, in lens systems it is responsible 
for the serious defect of chromatic aberration. 
In a simple lens, short waves are refracted more than long and 



































J 



(a) (b) (c) 

Fio. 6-16. Illustrating distortion, (a) The undistorted image of a square 
lattice; (6) the same image with "pin-cushion" distortion present; (c) the same 
image with "barrel-shaped" distortion. 




Fio. 6-17. Photographs to correspond to Fig. 6-16. 

will therefore be brought to a focus nearer the lens as shown in 
Fig. 6-18. This variation of focal position with wave-length is 
chromatic aberration. 

An ordinary uncorrected lens possesses this fault to a marked 
degree, shown in Fig. 6-19. This illustration was made in the 
same manner as that shown in Fig. 6-26, except that the light 




Fio. 6-18. Illustrating chromatic aberration. 

of a mercury arc was used as a source instead of light of a single 
wave-length. The separate rings of images of the source owe 
their positions to spherical aberration, but for each hole jn the 



SBC. 6-13J ABERRATIONS IN OPTICAL SYSTEMS , 



63 



screen, except the center one, a small spectrum is formed. There 
is a small amount of dispersion in the central image, since the 
hole at the center of the screen is not vanishingly small. Like- 
wise, for each other hole, there is a small amount of spherical 
aberration, which results in a blurring of the spectrum so that the 
separate images of the mercury spectrum have a tendency to 
overlap. For each hole in the ring nearest the center, however, 
the SDectrum is distinct. 




FIG. 6-19. Showing both spherical and chromatic aberration of a single lens. 
The screen shown in Fig. 6-2a was placed over the lens and the photographic 
plate placed at the Gaussian image point. Since the unfiltered mercury arc was 
used, each out-of-focus image is a spectrum. 

13. Cauchy's Dispersion Formula. The index of refraction 
of a transparent substance may be represented with sufficient 
accuracy for many purposes by Cauchy's formula 

n - n + + + - - , (6-16) 

\a2^4 * ^ " 

in which no, B t C, etc., are constants depending on the substance. 
For practical purposes it is sufficient to use only the first two 
terms of the right-hand side of eq. 6-16. 



64 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VI 



14. The Fraunhofer Lines. Accurate knowledge of indices of 
refraction of glass dates from the time of Fraunhojer, who was 
the first to measure the indices in terms of definite spectral 
positions instead of colors. He utilized the positions of the 
strong absorption lines in the solar spectrum, whose wave-lengths 
he found to be constant. His designations of these lines by 
letters are still used in optics. Since the development of strong 
laboratory sources of light, other reference lines have come into 
use, to most of which small letters have been assigned. In the 
following table a number of wave-lengths are given, including all 
the principal Fraunhofer lines. The unit used is the angstrom, 
equal to 10~ 8 cm. 



Wave-length, 
angstroms 


Due to 
element 


Approximate 
color 


Designation 


7685 


K 


Deep red 


A' 


7630 


O 


Deep red 


A 


6870 


O 


Deep red 


B 


6563 


H 


Red 


C 


5893 


Na 


Orange 


D 


5876 


He 


Yellow 


d 


5461 


Hg 


Green 


e 


5270 


Fe 


Green 


E 


4861 


H 


Blue 


F 


4359 


Hg 


Blue 


ff 


4340 


H 


Violet 


G 


4047 


Hg 


Violet 


h 


3968 


Ca 


Deep violet 


H 


3934 


Ca 


Deep violet 


K 



The variation of index of refraction with wave-length is small 
compared to the index itself. For ordinary glass it is never more 
than 2 per cent for the visible spectrum, i.e., for the range of 
wave-length represented in the table above, and it is frequently 
less. In designating a particular kind of glass it is customary 
among manufacturers to give as the principal means of identifica- 
tion the index of refraction for the D-line of Fraunhofer, and to 
add for working purposes the indices for several other lines, and 
the dispersive power, defined in Sec. 6-16. 

15. Two Kinds of Chromatism. By the term chromatic 
aberration is usually meant the difference with color of image- 
position distance from the lens. Even if a system is corrected 



SEC. 6-16] ABERRATIONS IN OPTICAL SYSTEMS 65 

for this defect, there might still be chromatism present, for, 
especially i the lens is thick, the principal planes for different 
colors will not necessarily coincide. The result will be a differ- 
ence of focal length for different wave-lengths, giving rise to a 
difference of magnification. This defect is known as chromatic 
difference of magnification, and sometimes as lateral chromatism. 
Difference of image position for different wave-lengths is known 
as axial, or longitudinal, chromatic aberration. 

16. Achromatizing of a Thin Lens. The focal length of a thin 
lens is given by 




= (n - l)k, (6-17) 

in which k is a constant for a given lens. By differentiation, 

df i j an w 

- - 



where the quantity co = dn/(n 1) is called the dispersive power. 1 
For a range of wave-length from the C- to the (7-lines, for instance, 
it may be written 

fl'Q ^C / r* t f\\ 

w = lr - T . (6-19) 

It should be pointed out that the numerator in eq 6-19 is not 
strictly an infinitesimal dn but a finite Aw. In other words, co 
is not the dispersive power for a particular wave-length, but the 
average dispersive power over a range of wave-length. The 
use of the symbol, dn, is justified by the fact that the difference 
of index over the visible spectrum is rarely more than about 
2 per cent of the index itself. 

For a lens made of twa_thin4eesin contact. 



1 = I + I, 
from which, by differentiation, is obtained 



_ 
/* 

<0l . C02 
= + , 

f > J\ h 

1 It is customary for glass makers to give the value of 1 /, sometimes 
called the Abbe number v. 



66 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VI 



since eq. 6-18 applies to each component separately. If the 
system is to be corrected for chromatic aberration, df/f* must be 
zero, and therefore 



= o. 

J2 



(6-21) 



Hence f\ and / 2 must be of opposite sign, since for all transparent 
substances <oi and W2 have the same sign. 1 Equation 6-21 says 
that achromatism is obtained by combining two thin lenses, one 
convergent and one divergent, of different dispersive powers. 
Their focal lengths may now be calculated. 
From eqs. 6-20 and 6-21, 



= 



O>2 



and 



/, 
2 = 



(6-22) 



Using eqs. 6-19 to 6-22, it is possible to calculate the correction 
over any desired range. First must be decided for what wave- 
lengths equality of focal length is desired. It is also important 
to notice that if the first lens is to be convergent and the com- 
bination also convergent, by eq. 6-22, f\ < /2. Hence, by 
eq. 6-21, <oi < o>2. 

A common combination Is a convergent lens of crown glass, 
and a divergent lens of flint glass, corrected for equality of focus 
for the F- and C-lines. Representative glasses of this type have 
the following indices : 





1. Crown 


2. Flint 


nc 


1.52293 


1.61549 


no 


1 . 52541 


1.62036 


Up 


1 . 53162 


1 . 63265 


no 


1 . 53652 


1.64307 



To calculate the focal lengths of the two lenses so that the 
focal length f D of the combination is 50 cm., we may proceed as 
follows : From eq. 6-19, 

, , n w - me 1.53162 - 1.52293 ftft1AWQf; 
^(crown) = ^_ 1 g-gggjj - 0.0165395. 

1.63265 - 1.61549 



co 2 (flint) = 



= 0.0276614. 



n ZD - 1 0.62036 

1 That is. in the Cauchv formula, the constant B is alwavs oositive. 



SEC. 6-16] ABERRATIONS IN OPTICAL SYSTEMS 67 

By eq. 6-22, 



fa = 'J^ m = 20.1037, 

CU2 



and 



To check, 

I-JL-l-JLs: J 1 = 1 

ID fio fa 20.1037 33.6223 50.0000* 
By eq. 6-17 

fc no 1 



n c 
so that 

fie = ; j- 

f _ J1D\n<lD x/ QQ QCOQ. 

J2C i d*5.5<5, 

and 

Jo- 



JIC "T /2C 

By a similar procedure it is found that 

f f = 50.0030 and f = 50.0925. 

The differencevS between / c , f D , and f F are negligible, but f is 
almost 1 mm. larger than either. This departure for wave- 
lengths outside of the range C F results in a diffuse circular 
area of color about an image point, which is known as a secondary 
spectrum. 

The radii of curvature of the lens surfaces may be found if the 
shape of one lens is decided upon. The choice of radii is ordi- 
narily such as to reduce other aberrations to a minimum. A 
common form of achromat is an equiconvex lens of crown with a 
divergent lens of flint glass cemented to it. Let n = r 2 for 
the convergent lens. Then the first surface of the divergent lens 



68 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VI 



will have the same radius as r 2 . Using the index for the D-line 
and applying eq. 6-17 in turn for each of the lenses, we obtain 

n = 22.1061, r z = -22.1061, 
r, = -22.1061, r, = -369.402. 

It is important to point out that the achromatism thus obtained 
is an equality of focal length only, unless the lenses are very thin ; 
also that combining two lenses of different indices does not 
accurately achromatize the combination for more than two wave- 
lengths. In order to make a system achromatic or nearly so for 
any appreciable region of the spectrum it is necessary to use more 
than two elements. For the lens to be achromatic with respect 
to image position, each element must be separately achromatized. 
This is because the focal planes are not the same for different 



a 




FIG. 6-20. 

wave-lengths. In Fig. 6-20 the lateral magnification of the first 
lens is y\/y = a' /a, that of the second is y'/yi = b'/b, and that 
of the whole system is y'/y = b'a'/ba. In order that the system 
may be achromatized with respect to image position, there must 
be no difference of the distance b' for different wave-lengths, or 



A6' = 0, 



(6-230), 



and for constancy of lateral magnification for different wave- 
lengths, 



(6-236) 



Since a is constant, the condition in eq. 6-236 may be written 
A(6'a'/6) = 0. But 6 -f- a' is constant for all wave-lengths, 
hence 




Aa' = -A6. 



(6-24) 



SBC. 6-18] ABERRATIONS IN OPTICAL SYSTEMS 69 

By eqs. 6-23a and 6-236, A(6/a') = and A6 = 0. This means 
that each of the lenses must be achromatized separately in order 
that the combination may be achromatic both with respect to 
focal length and to focal plane. The condition a = constant 
means that the system will be achromatic for only one position 
of the object. Usually a combination is corrected for the object 
at infinity. For objects nearer to the lens the achromatism will 
be sufficient for most purposes. ) 

17. The Huygens Ocular. It is possible to arrange a com- 
bination of two thin lenses in such a manner that a high degree 
of achromatism is attained, even though the two are of the same 
kind of glass. For thin lenses separated by a distance t, 

1 _ 1 1 t 

7" i T 



f fl f* /!/ 

Differentiating, 



/ 2 /I 2 /2 2 V /I/'/ 2 

But co = TJ so 



df _ 0>i , C02 (CO) + CO->)/ 

~ " ' 




If the combination is achromatic with respect to focal length, 
this must be zero, i.e., 

. _ t02/l + tO 1/2 
COi + C02 

If 2 = wi, i>e., if the elements are of the same kind of glass, 

t = ^4^ 2 - (6-25) 

^J 



Thus, if two thin lenses are placed a distance apart equal to half 
the sum of their focal lengths, the combination is achromatic with 
respect to focal length for all colors, but it possesses bad axial 
chromatic aberration. 

18. The secondary spectrum can be reduced with two lens 
elements of different indices of refraction, provided the lens 
having the higher index has the smaller dispersive power. This 



70 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VI 

is true of glasses developed at the Jena Glass Works. With 
these glasses it is also possible to correct for chromatic aberration 
and curvature of the field at the same time. The Petzval con- 
dition for no curvature of the field may be written 



= 0, 
while eq. 6-21 may be written 

/i2 +/2Wi = 0; 
eliminating /i and / 2 , the result is 



This result holds, it must be remembered, only for lens systems 
which are so thin that no variation of the position of the principal 
planes with color exists. Actually, very good achromats which 
are not ideally thin can be made. A system which is aplanatic 
and achromatic for two or more colors, and is free from secondary 
spectrum is called apochromatic. 

It is clear that there can be no such thing as perfect image 
formation such as is postulated by the theory of ideal optical 
systems. Some of the aberrations cannot be entirely eliminated, 
and it is possible only to reduce them to a degree consistent with 
the purpose for which the system is intended. The practical 
lens maker accomplishes this by tracing representative rays 
through the system. 1 

Problems 

1. The radii of curvature of both faces of a thin convergent lens are 
the same length. Show that for an object placed a distance from the 
lens equal to twice its focal length the longitudinal spherical aberration 
is given by 



2/(n - 1)2' 

* 

2. A spherical wave from a near-by source is refracted at a plane 
surface of glass. What will be the character of the wave front after 
refraction? Will it be free from aberrations? 

3. Find the lengths and positions of the astigmatic line images formed 
by a concave mirror whose diameter is 10 cm. and whose radius of 

1 For an exposition of these methods the student is referred to Hardy and 
Perrin, "The Principles of Optics," McGraw-Hill. 



ABERRATIONS IN OPTICAL SYSTEMS 71 

curvature is 50 cm. if the source is a point 75 cm. from the axis on a 
plane 125 cm. from the vertex of the mirror. Find also the astigmatic 
difference. 

4. Using Young's construction, show the path of a ray refracted at a 
convex lens surface of radius +r, (n'<n). 

5. Locate the conjugate aplanatic points of a spherical glass refracting 
surface of radius +5, if the index is 1.57. 

6. Calculate the constants of the doublet described in Sec. 6-16, if it 
is achromatized for the C- and (r-lines, instead of for the C- and F-lines. 



CHAPTER VII 

OPTICAL INSTRUMENTS 

1. The Simple Microscope. The Magnifier. If an object 
is held somewhat closer to a thin positive lens than its principal 
focal point and viewed through the lens, an enlarged, erect, 
virtual image will be seen. Used in this way, the lens is a simple 
magnifier. Its magnification is the ratio of the size of the image 
formed on the retina with the aid of the lens to its size when viewed 
by the unaided eye at normal reading distance. If this distance 
is called N, and the virtual image formed by the magnifier is 
considered to be N cm. away from the eye, then by eq. 2-7 the 
lateral magnification is 

R _ tf _ N + E 
V- r -- r 

But E, the distance between the emergent focal point F f of the 
magnifier and the eye, is usually very small compared to TV and 
may be neglected. Also, it is customary to consider N to be 
about 25 cm., so that the magnification of a simple magnifier 
may be written 

25 

/3 = -y- (/in centimeters). (7-1) 

Here/ is used instead of/', since for a lens in air they are the same. 
It is best to avoid eyestrain by placing the object at the principal 
focus F of the lens (see Fig. 7-1) so that the virtual image is at 
infinity. This does not invalidate eq. 7-1, as a good working 
rule, since the angle subtended by the virtual image at infinity 
is not much different from that at normal reading distance, and 
the virtual image is about the same size. This may be quickly 
verified by experiment. To obtain the largest field, the eye 
should be close to the lens. A simple magnifier may be corrected 
in the usual manner for chromatic aberration. 

2. Compound Magnifiers* Because large magnification causes 
great increase of the aberrations, simple magnifiers are usually 

72 



SEC. 7-2] OPTICAL INSTRUMENTS 73 

limited to magnifications smaller than about 15. Compound 
magnifiers usually consist of two lenses. One type of compound 
magnifier is the Ramsden eyepiece, ordinarily used as an ocular 
in a telescope or microscope. As shown in Fig. 7-12, page 83, 
it is made of two plano-convex lenses with their convex surfaces 
toward each other. Two thin lenses thus used, it was shown in 
Chap. VI, form a combination which is achromatic with respect 
to focal length, provided the distance between the lenses is one- 

From 
image 




o 



FIG. 7-1. The simple microscope. 

half the sum of their focal lengths, and provided they are made 
of the same kind of glass. There is, however, always some axial 
chromatic aberration present, and on this account the focal lengths 
are calculated for the yellow green (about 5500 angstroms), to 
which the eye has maximum sensitivity. An eyepiece thus con- 
structed will, however, have its incident focal plane at the first 
surface of the field lens, x and dirt or surface imperfections of that 
lens will be in sharp focus. Consequently, 
at some sacrifice of achromatism the distance 
between the lenses is made two-thirds the 
focal length of either, instead of one-half the 
sum of the focal lengths. Fl0 :. 7 ; 2 -- The . 

dmgton eyepiece. 

The Coddington eyepiece (Fig. 7-2) is made 
of a single piece of glass cut from a sphere, with a groove cut in 
its sides to form a stop. Loss of light by reflection between sur- 
faces is reduced by this eyepiece, but it is expensive to make. 

The triple aplanat (Fig. 7-3) is made of two negative lenses of 
flint glass, between which is cemented a double-convex lens of 
crown glass. In this magnifier a high degree of achromatism is 
attained. 

1 The field lens is the one closer to the focal plane of the objective of the 
telescope or microscope. 




74 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VII 




There are several other types of magnifiers, most of which are 
constructed for special purposes. One of these is a single block 
of glass with a spherical top and with a flat or slightly concave 
base to be placed in contact with the object. 

3. The Gauss Eyepiece. For laboratory telescopes, espe- 
cially those used with spectrometers, the Gauss eyepiece is 
very convenient. Its construction is shown in Fig. 
7-4. It is the same as the Ramsden eyepiece, 
~~" except that between the two elements a thin plane 
plate of glass is placed at an angle of 45 deg. with 
FIG. 7-3- The the axis. Light admitted through an opening in 
rip e ap ana . ^ e g^ Q f j. ne fafe i s re fl ec tcd down the axis of the 

telescope, illuminating the cross hairs in its path. When the 
telescope is focused for parallel light and has its axis perpendic- 
ular to 'a plane reflecting surface placed before the objective, 
images of the cross hair will be at the principal focus of the 
objective. When these images coincide exactly with the cross 
hairs themselves, the axis of the telescope is exactly perpendicu- 

Source 



Cross- 
hairs 



U 



J 




Draw fube- f 



FIG. 7-4. The Gauss eyepiece and draw tube. 

lar to the reflecting surface. The focusing of the telescope for 
parallel light may also be made more exact by eliminating all 
difference of sharpness between the cross hairs and their images. l 
4. The Micrometer Eyepiece. When small distances are to 
be measured, a convenient instrument is the micrometer eyepiece. 
This can be constructed in several forms, one of which is illus- 
trated in Fig. 7-5. At the focal plane of the eyepiece are a fixed 



1 For instructions concerning the use of the Gauss eyepiece, see 
Appendix IV. 



SBC. 7-6] 



OPTICAL INSTRUMENTS 



75 



cross hair F, and a movable cross hair M . The latter may be 
moved perpendicularly to the axis of the eyepiece by means of a 
fine-pitched screw. The head H may be divided into appropriate 
divisions, usually small fractions of a millimeter, although for 



1 '' ' p 
\ 
\ 


!l!!l 


X 

\ 
\ 
\ 

,1 I 

III',' I 






-J 

"^> 


E 9 
-TO 




~ 




J- 


H 


F M 

FIG. 7-5. 



some purposes angular measure is more convenient. With a 
head of sufficient diameter the divisions may represent very small 
distances or fractions of a degree, and in addition a vernier may 
be used. It is not practical, however, to make divisions smaller 
than are justified by the accuracy of the micrometer screw. 
Sometimes a small-toothed edge is provided in the focal plane 
so that whole turns of the micrometer head may be easily counted. 
If the ocular is of the Huygcns type (see Sec. 7-12), the cross hairs 
are placed at the focal plane of the eye lens, and for the toothed 
edge may be substituted a scale finely ruled on glass. 



Objective 




- Image at 
r infinity 
IG. 7-6. The compound microscope. 

6. The Compound Microscope. The optical parts of a com- 
pound microscope consist of an objective and an eyepiece or ocular. 
The former serves to produce a much enlarged real image of the 
object; the latter, to view this image with still further magnifica- 



76 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VI I 

tion, in comparison with a scale if measurements of the object 
are to be made. A schematic diagram is shown in Fig. 7-6. 
The magnification is the product of the magnifying powers of 
the two elements. Frequently a microscope is equipped with a 
variety of objectives and oculars for different magnifications. 

6. Numerical Aperture. In the chapter on diffraction it will 
be shown that the radius of the image of a point object, i.e., the 
distance from the center of the image to the first minimum of t ho 
diffraction pattern, is given by 




r = , (7-2) 



where is the angle of diffraction, i.e., the angle subtended at the 
lens by r. By convention, the limit of resolution of an optical 
instrument is said to be reached when the center of the image of 
one object just coincides with the first dark minimum of the 
diffraction pattern of a second object. Hence images of two 
point objects can just be resolved when their distance apart is r. 

In the microscope, on account of the greater magnification, 
the angle is large. Moreover, the object points seen with a 
microscope are not self-luminous, and hence in themselves pro- 
duce diffraction images of the source. Abbe has shown that in 
consequence the smallest distance between two points in the 
object which can be resolved is given by X/2n sin a, where n is 
the index of refraction of the medium between the object and the 
objective, and a is the angle between the axis and the limiting 
rays which pass through the entrance pupil of the microscope. 
The quantity n sin a was called by Abbe the numerical aperture 
(N.A.) of a microscope. It is obvious that the limitation thus 
set on the magnifying power is not due to aberrations but to the 
effects of diffraction. From eq. 7-2 it follows that the size of the 
central bright maximum of the diffraction pattern of a point 
object is proportional to the wave-length of the light used. For 
this reason, sometimes ultraviolet light is used to obtain higher 
resolving power. 

The oil-immersion microscope 1 is one in which the numerical 
aperture and hence the resolving power is increased by the use of 
an oil, usually oil of cedarwood, between the object and the 
objective. Loss of light by reflection is thereby also eliminated. 

1 See Sec. 6-9. 



SEC. 7-8J 



OPTICAL INSTRUMENTS 



77 





7. Condensers. If the object is viewed with transmitted 
light, it is frequently desirable to obtain greater illumination by 
* "^ans of a condenser. Sometimes this is merely a concave 
mirror, which can be adjusted to reflect a convergent beam of 
light from a nearby source into the objective. For short-focus 
objectives more powerful condensers are used. They become in 
fact integral parts of the optical system, and are often corrected 
for aberrations so as to improve their light-gathering power. 
The larger the numerical aper- 
ture, the more important does 

the efficiency of the condenser 
become. 

8. Vertical and Dark-field 
Illuminators. When very 
short-focus objectives are used 
to view opaque objects which 
must be illuminated from 
above, it is difficult to illumi- 
nate the object by ordinary 
means. To overcome this diffi- 
culty, a vertical illuminator may be used. This may consist of a 
prism or mirror which reflects to the object a beam of light 
admitted into the tube from the side, as shown in Fig. 7-7. 

For observing small particles in colloidal suspensions, or fine 
rulings on surfaces, it is desirable to use a dark-field illuminator. 
In this type the light is incident upon the object at angles such 
that it does not pass by transmission or ordinary specular 1 
reflection directly into the objective. Small particles or lines, 
however, serve to diffract the light, and it is by means of the 
pencil of diffracted light from each particle that the presence 
of the particle is observed. One means of effecting this is by 
means of condensers such as are illustrated in Fig. 7-8. The 
condenser contains an opaque centered disk which allows only a 
ring of light to pass obliquely through a point in the object 
just below the center of the objective. With dark-field illumina- 
tion, particles as small as 5 X 10~ 7 cm. in diameter, or about 
of the wave-length of light, may be observed. 



<cO (b) 

Fi. 7-7. Vertical illuminators. 



1 Specular reflection is ordinary reflection from a polished surface, diffuse 
reflection from a matt surface. 



78 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VII 



9. Telescopes. In Chap. Ill it was shown that the lateral 
magnification of a telescope is given by ft; the negative sign 



indicating that the image is inverted. The angular magnifica- 
tion is given by fi/fz- The latter is commonly spoken of as the 
magnifying power of the telescope. Rigorously, a telescopic 
system is one which forms at infinity an image of an infinitely 
distant object. In practice the term telescope is also applied to 
any instrument used for forming images of nearby objects, or for 





Av -'-4 



1"^ 


^\ 


1 
1 


A* \ 


1 

1 


\ 
I 


1 

i 


1 
I 




(a) 

FIG. 7-8. (a) The Abbe condenser. A is an opaque screen; (', the condenser 
system; O, the object, (b) The Cardioid condenser. A is the opaque screen; 
*S', a spherical reflector; C, the cardioid surface; O, the object. At points / are 
layers of oil. 

forming images at finite distances. An example of the first- 
mentioned use is the ordinary laboratory telescope, used for 
observing objects a few feet away. If it is used to observe objects 
closer than the normal reading distance of 25 cm., such an 
instrument is called, instead, a microscope. Optically, for 
such ranges of distance, there is little difference between a short- 
range telescope and a long-focus microscope. 

The modern astronomical telescope is used principally for 
photographic purposes; it consists of a single lens or mirror for 
focusing images of celestial objects on the photographic plate. 
The modern telescope is thus principally an instrument of great 
light-gathering power. From the laws of diffraction it can be 



SEC. 7-10] OPTICAL INSTRUMENTS 79 

shown that the resolving power of a telescope is determined by the 
size of the objective. To take full advantage of this, however, 
in visual use, it is necessary that the objective be the aperture 
stop of the system. It is then the entrance pupil. In a visual 
astronomical telescope in which both objective and ocular are 
positive systems, a measure of the magnifying power may be 
made by comparing the diameter of the objective with that of the 
exit pupil, since the ratio of these two dimensions is equal to 
/i//2- To find the size of the exit pupil, the telescope may be 
pointed to the sky and a ground glass or paper screen used to 
locate the position where the beam emerging from the eyepiece is 
smallest. The well-defined disk of light at this point is the exit 
pupil. 

In order that the maximum field may be viewed, the entrance 
pupil of the eye should be made to coincide with the exit pupil of 
the telescope. In the Galilean telescope, the exit pupil is virtual, 
and the field of view is in consequence restricted. This form 
has, however, the advantage of shorter overall length, since the 
eyepiece is a negative lens placed closer to the objective than 
its principal focal point. 

10. The Reflecting Telescope. Large modern astronomical 
telescopes which are used principally for photographic observa- 
tions are of the reflecting type. The mirror is a parabolized 
surface, usually of glass coated with metal of high reflecting 
power. Silver, chemically deposited, was until recently the 
metal used. The disadvantage of silver is that it tarnishes 
readily and loses its reflecting power. With recent improvement 
in technique it is now possible to deposit aluminum by evapora- 
tion in a high vacuum on even the largest mirrors. The oxide 
formed on the aluminum on exposure to the air is an extremely 
thin coat of transparent substance preserving the metal from 
tarnish. Sometimes combinations of two metals prove more 
satisfactory than aluminum alone, as, fir instance, a base coat 
of chromium with a top coat of aluminum. Indeed, the tech- 
nique of evaporation of metals for the production of reflecting 
surfaces is so new that probably great improvements will be made 
in the future. In addition to its value in the visible spectrum 
because of superior reflecting power and durability, aluminum 
has proved of great service in extending astronomical spectro- 
scopic observations into the ultraviolet. Silver is almost trans- 



80 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VIJ 

parent in the region of 3300 angstroms, a fact which previously 
limited the ultraviolet spectroscopy of stellar objects. 

The image formed with a paraboloidal mirror is free from 
spherical aberration on the axis and from chromatic aberration 
over the entire field. For this reason, it is possible to make the 
focal length shorter with relation to the aperture than in a 
refracting telescope. The new telescope of the McDonald 
observatory has a diameter of 82 in. and a focal length of 26 ft., 
or a relative aperture of //3.8, resulting in a reduction of photo- 
graphic exposure time. In this way, the range of the instrument 
for faint celestial objects is effectively increased. 

In the neighborhood of the axis, the field of a paraboloidal 
reflecting telescope suffers from coma, a defect in which each 
image is elongated as shown in Fig. 6-9. The length of the 
comatic image, i.e., its dimension measured along a radius from 
the center of the field, is given approximately by L = 3/> 2 0/16F 2 , 
where D/F is the relative aperture and 6 is the angular distance 
of the star image in seconds of arc from the center of the field. 
The breadth of the comatic image is approximately two-thirds of 
the length. While the defect of coma thus increases in propor- 
tion to 0, the astigmatism is proportional to 2 , so that for tho 
region close to the axis the elimination of coma is more important. 
In actual practice, differences of temperature in a turbulent- 
atmosphere cause a blurring or " boiling," so that even stellar 
images at the axis are enlarged and irregular. For this reason 
the distance from the axis at which coma becomes noticeable 
depends upon the definition, or " seeing," as it is called. For 
correcting this defect of coma the telescopes which have been 
developed may be classified in three groups. 

a. In the first group may be placed those which achieve their 
purpose to some extent by the addition of other mirror surfaces, 
which may or may not be modified from a spherical shape. 
While the original two-surface reflecting telescope proposed by 
Gregory (Fig. 7-9a) and the Cassegrain form (Fig. 7-96) fall 
in this group, the greatest advance was made by Schwarzschild 
who, in 1905, designed a two-mirror telescope of the Gregorian 
type in which each surface was modified in shape so that coma 
and spherical aberration were reduced to a minimum in the 
neighborhood of the axis. It had a relative aperture of 1/3.5. 
Since in this instrument the field is flat and the residual astigma- 



SBC. 7- JO] 



OPTICAL INSTRUMENTS 



81 



tism is balanced so that the primary and secondary foci coincide 
at short distances from the axis, it fulfills the conditions for the 
anastigmat described in Sec. 6-6, and can be called the Schwarzs- 
child anastigmat. 

b. In a second group may be placed those telescopes in which 
the coma of a paraboloidal mirror is corrected by the use of a 
specially designed lens placed between the mirror and its principal 
focus. The disadvantage of correcting lenses of this type is that 
in many cases they reduce the relative aperture by increasing the 
focal length. 



Oregorian 




Newtonian 




(c) 
Fio. 7-9. -Early types of reflecting telescopes. 

Professor Frank E. Ross of the Yerkes Observatory has 
designed a "zero-power" lens combination 1 placed between the 
mirror and its focal point, which makes no essential change 
in the position of the principal focal plane of the telescope mirror, 
but corrects for coma over a considerable area. It makes no 
reduction in relative aperture, and actually increases the photo- 
graphic speed of the telescope since the comatic images are 
decreased in size. 

c. A third type of correcting device is a single plate which has 
surfaces so shaped that it modifies the character of the bundle 
of rays from a point source before it reaches the reflecting mirror. 
A most successful corrector of this type designed by B. Schmidt 

1 Astrophysical Journal, 81, 156, 1935. 



82 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VII 



is illustrated in Fig. 7-10. The reflector is made spherical instead 
of paraboloidal, so that coma is absent. At its center of curva- 
ture is placed a disk of glass, one of whose surfaces is plane and 
the other shaped so that the light refracted by it is made slightly 
divergent by an amount necessary to eliminate spherical aberra- 
tion. With a telescope of this type in which the mirror has a 
diameter of 71 cm. and a focal length of 1 m. good star images 
have been obtained over a field 12 degrees in diameter. 





FIG. 7-10. (a) An ordinary spherical reflector; (6) the Schmidt reflecting 
telescope. The curvature of the upper surface of the compensating plate is 

exaggerated. 

/ 

11. Oculars (Eyepieces). In the section on magnifiers it was 
pointed out that a Ramsden eyepiece makes an excellent reading 
glass or magnifier. --In fact the only difference between magni- 
fiers and oculars is that while the former are used to view real 
objects, the latter are used to view images formed by another 
part of an optical system. Any magnifier will serve as an 
eyepiece for a telescope or microscope, but most of them not so 
well as an eyepiece specially constructed for the purpose. The 
triple aplanat in particular makes an excellent eyepiece. 

12. The Huygens Eyepiece. The two principal types of 
oculars are the Huygens and the Ramsden. ^ The Huygens, 
sometimes called a negative ocular, is illustrated in Fig. 7-11. It is 
made of two elements of the same kind of glass separated by a 



SEC. 7-13] 



OPTICAL INSTRUMENTS 



83 



distance equal to one-half the sum of their focal lengths. The 
field lens A is placed just inside the focus F of the objective, 
this focus serving for the field lens as a virtual object of which an 
erect image I is formed closer to A. The eye lens B is so placed 
that / is at its focal point, thus forming an image at infinity. 
The ratio of the focal length of the field lens to that of the eye 
lens is about 2:1 if the eyepiece is to be used for a microscope and 
somewhat larger for a\^telescope. Sometimes a scale or cross 



/ F 




-The Huygens eyepiece. 



hairs, or both, are placed at /, but if the eye lens is uncorrected 
for aberrations, the scale cannot be very long. Because the 
distance between the elements is (/i /2)/2, the Huygens ocular 
is free from chromatic aberration with respect to focal length, 
although the longitudinal aberration and curvature of the field 
are considerable. These may be corrected by special means, 
such as changing the curvature of the surfaces of the field lens 




Field Field Eye Exit 

stop lens lens pupil 

FIG. 7-12. The Ramsden eyepiece. 

while retaining its converging power, or achromatizing the 
eye lens. 

fer 13. ^The Ramsden Eyepiece. The essentials of construction 
of the Ramsden eyepiece, shown in Fig. 7-12, have been described 
in the section on magnifiers. This ocular has a flatter field than 
the Huygens and possesses the added advantage that the focal 
plane of the objective precedes the field lens, so that a scale or 



84 



LIGHT: PRINCIPLED AND EXPERIMENTS [CHAP. VII 



traveling cross hair can be used more successfully. The Ramsden 
also has the added advantage that it may be focused very sharply 
on the cross hairs or scale. This is important if eyestrain is to be 
avoided. The observer should first relax the accommodation by 
resting the eye on a distant view; then, while looking in the eye- 
piece, he should draw it away from the cross hairs until they 
just begin to appear diffuse. Two or three trials will quickly 
determine the correct focal position for the eyepiece. 

14. Erecting the Image. Sometimes it is desirable to have 
an erect image instead of the inverted image seen in the ordinary 



Erect virtual 
image 




(a) 




Inverted -^T- ^ 
virtual image"' 

Fio. 7-13. (a) The Galilean refracting telescope; (6) the astronomical refracting 

telescope. 



eyepiece. In the prism binocular this is done by means of 
prismatic reflections. The simple negative lens of the Galilean 
telescope (Fig. 7-13a) also serves to erect the image. In terres- 
trial telescopes, where it is desirable that a distant scene be 
erected, a four-element eyepiece, illustrated in Fig. 7-14 is used. 
15. The Spectrometer. Perhaps the most important optical 
instrument for the study of light is the spectrometer. It may be 
used to determine indices of refraction, to study the effects of 
diffraction, interference, and polarization, and to make observa- 
tions on spectra. For the last-named purpose it has reached its 
greatest development in the spectrograph which is essentially a 
spectrometer equipped with a camera in place of the eyepiece. 



SBC. 7-15] 



OPTICAL INSTRUMENTS 



85 



The essential parts of a spectrometer are shown in Fig. 7-15. 
At S is a slit, with accurately parallel jaws, which may be altered 
in width from about 0.001 mm. to a few millimeters. The varia- 
tion in width may be accomplished by a motion of one jaw 
(unilateral), if only narrow slits are to be used, or of both jaws 
equally (bilateral) in case wide apertures are desired, or sym- 




FIG. 7-14. -The erecting eyepiece. 

metry of widening is to be maintained. The slit is mounted at 
one end of a tube, at the other end of which is the collimator 
lens Li which for ordinary visual purposes must be a good crown- 
flint achromat. The collimator tube is equipped with one or 
more devices for altering its length. Usually this is accomplished 
by a rack and pinion which can be turned to change the slit 
distance from the lens. At L 2 is a second lens, preferably an 




FIG. 7-15. The spectrometer. 

achromat identical with L\. This lens, the tube on which it is 
mounted, and the eyepiece E constitute a telescope. At the 
focal plane of the objective are mounted the cross hairs. The 
distance of the cross hairs and eyepiece from the objective may 
be changed by means of a rack and pinion. In some cases, the 
lenses L\ and L 2 are also independently mounted on drawtubes 



86 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VII 

which may be clamped to the main tubes at different distances. 
The collimator and telescope tubes are mounted on arms which 
have as an axis of rotation the central vertical axis of the spec- 
trometer. With this axis as a center of rotation also are the 
spectrometer table T on which prisms or other optical parts may 
be mounted between the collimator and telescope, and a grad- 
uated circular scale. The circular scale should be as large as the 
dimensions of the instrument will permit. It is divided into 
units of angle, and may be graduated from to 180 deg. in two 
equal sections or from to 90 deg. in each quadrant. It should 
be read at opposite sides, either by means of verniers or by 
microscopes. In addition to being adjustable about the central 
axis, the collimator and telescope are also adjustable about 
vertical axes and horizontal axes perpendicular to their lengths. 
When the instrument is in adjustment, the longitudinal axes of 
the collimator and telescope should meet at the central vertical 
axis of the spectrometer. 

Directions for the adjustment of a spectrometer will be found 
in Appendix IV. 

Problems 

1. What is the magnifying power of a glass ball 1.5 cm. in diameter? 
(n = 1.5) 

2. A piece of capillary glass tubing has an outside diameter of 7 mm. 
The capillary appears to be about 1 mm. wide when looked at through 
the glass wall. What is its real diameter? (n 1.5) 

3. A magnifying glass whose focal length is 6 cm. is used to view an 
object by a person whose smallest distance of distinct vision is 25 cm. 
If he holds the glass close to the eye, what is the best position of the 
object? 

4. The objective of a telescope has a diameter of 30 mm. and a focal 
length of 20 cm. When focused on a distant object, it is found that the 
diameter of the exit pupil is 2.5 mm. What is the magnifying power 
of the system? If the eyepiece is a single thin lens what is its focal 
length? 

6. A celestial telescope has a focal length of 25 ft. What must be the 
focal length of an eyepiece which will give a magnification of 300 
diameters? 

6. The objective of a telescope has a focal length of 40 cm. and the 
ocular has a focal length of 5 cm. Plot the magnification as a function 
of object distance, if the latter varies from 5 m. to infinity. 



OPTICAL INSTRUMENTS 87 

7. The objective of a field glass has a focal length of 24 cm. When 
it is used to view an object 2 m. away the magnification is 3.5. What 
is the focal length of the ocular? What will be the magnification for an 
object a great distance away? 

8. A large astronomical telescope usually has a "finder 11 attached 
to it, which consists of a short-focus telescope fastened to the cylinder 
of the larger one. Explain the use of the finder. 



CHAPTER VIII 
THE PRISM AND PRISM INSTRUMENTS 

1. The Prism Spectrometer. Probably the most important 
use of the prism is for the spcctroscopic analysis of light. Because 
of the variation with wave-length of the index of glass and other 
transparent substances, light passed through a prism is spread 
out into a spectrum by means of which the analysis may be made. 




FIG. 8-1. ---A section through a prism perpendicular to the refracting edge. 

In Fig. 8-1 is shown a section made by passing a plane through a 
prism perpendicular to the two refracting surfaces and the 
refracting edge of the prism, i.e., the edge in which the refracting 
surfaces intersect. The plane of the paper is the plane of 
incidence. A beam of) light incident on the first surface is bent 
by refraction through an angle i r, and at the second surface 
through an angle i' r' , where i and i' are the angles made 
between the directions of the beam in air and the normals to the? 
surfaces. The total deviation is thus A = i + i' r r'. 
From the geometry of the figure it is easily proved that A, the 
refracting angle of the prism, is equal to r H- r', so that 

A - i + i' - A. (8-1) 

If the incident beam is fixed in direction, and the prism rotated 

88 



SEC. 8-2] THE PRISM AND PRISM INSTRUMENTS 89 

clockwise, i and r will increase, and r' and i' will decrease, while 
a counterclockwise rotation will cause i and r to decrease and 
i' and r' to increase. It can easily be shown by a simple experi- 
ment that the value of A for any wave-length will pass through 
a minimum as this rotation takes place. A necessary condition 
for this minimum is that the derivative of A with respect to i 
shall be zero, i.e., 



or 

gr = -1. (8-2) 

To evaluate this, the equations for Snell's law applied to the 
refractions at each surface may be differentiated, resulting in 

cos i di = n cos r dr, 
cos i' di' n cos r' dr'. (8-3) 

Then since A = r + r', 

dr = dr'. (8-4) 

It follows from eqs. 8-2, 8-3, and 8-4 that A is a minimum if 

cos i _ cos r 
cos i f cos r' 

i.e., if i = i'. Therefore at minimum deviation, i = i' and 
r = / = A/2; and by eq. 8-1, i = (A + A)/2. Substituting 
these values in Snell's law, we have for the index of refraction. 

'A + A' 

n = .\A/' (8-5) 

sin (A/2) 

Thus the index of refraction of a transparent substance may be 
found if it is cut into a prism for which the angle A and the 
minimum deviation A are measured. It should be noted that 
there will be a different angle of minimum deviation for every 
wave-length. 

2. Dispersion of a Prism. If the source is made a very narrow 
illuminated slit perpendicular to the plane of incidence, then the 



90 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VIII 

dispersion resulting from the dependence of n upon X will result 
in a spectrum in which each wave-length of the incident light will 
be a line, i.e., an image of the slit, 1 whose magnification depends 
upon the focal lengths and the adjustment of the telescope and 
collimator. 

Dispersion can be defined as the rate of change of deviation 
with wave-length. For a given prism angle A and a constant 
angle i the deviation changes with index of refraction and with 
i'. Thus the definition of dispersion just given becomes, from 
eq. 8-1, 

n dA _ di' 

v ~j\ ~j\" 
d\ d\ 

This expression is for the angular dispersion, and not for the 
actual separation of the spectral lines at the focal plane of the 
telescope objective. Its value cannot be obtained in a single 
step since there is no simple expression connecting i' and X. 
It is possible to find di' /dn from Snell's law and dn/d\ by differ- 
entiation of the Cauchy dispersion function of eq. 6-16; these 
multiplied together give the desired expression for di'/d\. This 
derivation will now be made. 

The Cauchy dispersion formula 

n - Wo + r + ' ' ' 

is an empirical relation in which the values of no and B are experi- 
mentally determined. It is not valid when the region of the 
spectrum to be considered contains absorption bands, but it is 
satisfactory for ordinary transparent substances such as glass, 
quartz, fluorite, etc. By differentiation, 

dn 2B , Q ^ 

3x = ~ v" (8 ' 7) 

In order to obtain di' /dn we may proceed as follows: From 
Snell's law, for the second surface, 

cos i r di' = dn sin r' + n cos / dr f , (8-8) 

1 If only a slit and prism are used, and the spectrum lines viewed with the 
unaided eye, the optical system of the eye produces the images of the slit. 



SEC. 8-2] fHE PRISM AND PRISM INSTRUMENTS 



91 



while for the first surface, since i is constant, 

n cos r dr + dn sin r = 0. 
Since dr dr', eq. 8-9 may be written 

7 , dn sin r 

dr = 

n cos r 

which, substituted in eq. 8-8, gives 

di' sin (r + /) sin A 



(8-9) 



dn cos i cos r cos z cos r 
Multiplying eqs. 8-7 and 8-10 gives for the dispersion 

di' -2B sin A 



(8-10) 



X 3 cos i cos r 
Thus the dispersion of a prism depends on four factors: 

A 

Pi 



(8-11) 
(1) the 




FIG. 8-2. 

character of the glass, given by the constant B, (2) the wave- 
length X, (3) the refracting angle A of the prism, and (4) the 
direction of the light through the prism as given by the angle r. 
The value of i' will, of course, depend on r, A, and B, and also on i. 
For the case of minimum deviation, eq. 8-11 may be simplified. 
Consider a beam of light of width a, composed of parallel rays, 
emerging from the prism. At minimum deviation (Fig. 8-2), 
a = P'Q' cos i', R'Q' - P'Q' sin (A/2), whence 



P'Q' = 



(QQ' - PP'} 

2 sin (A/2) ' 



92 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VIII 

If we call QQ' PP' the effective thickness t, then 

t cos i' 



a 



2 sin (A/2) 



Putting the value of cos i' from this equation in eq. 8-11, modified 
for the condition of minimum deviation, we obtain 

di f 2Bt 



The negative sign has been dropped since it merely indicates, as 
shown by eq. 8-7, that as X increases, n decreases. 

Each image of the slit, or spectrum line, is observed to be 
curved. Rays from the upper and lower ends of the slit undergo 
greater dispersion than those from the center, having a longer 
path through the prism because they pass through at an angle 
with the optical axis of the system, i.e., with the plane of Fig. 
8-1. The greater the distance above or below the axis is the part 
of the slit from which a ray comes, the greater is the dispersion, 
thus accounting for the fact that all lines are curved, with their 
ends pointing toward shorter wave-lengths. 

It should be emphasized that the dispersion D given by eqs. 
8-11 and 8-12 is the change of angle with wave-length. Some- 
times the term dispersion is used to mean the separation in 
angstroms per millimeter in the field of the telescope or on the 
photograph of the spectrum. Thus a spectrum is said to have a 
dispersion of 10 angstroms per millimeter when two spectrum 
lines whose difference of wave-length is 10 angstroms are just 
1 mm. apart. Obviously this quantity will depend not only 
on the angular dispersion D, but also on the focal length, i.e., 
on the magnification of the telescope or camera objective. 

3. Resolving Power of a Prism. It is customary to define 
the resolving power of a dispersive instrument as 

R = 1 



where d\ is the smallest wave-length difference which can be 
detected at the wave-length X. It will be shown in the chapter 
on diffraction that limitation of the wave front from a narrow- 
slit source by an aperture of width a results in an intensity 
distribution in the image, shown in Fig. 12-17. Moreover, two 



SEC. 8-3] THE PRISM AND PRISM INSTRUMENTS 93 

such images are said to be just resolved when the middle of one 
coincides with the first minimum of intensity of the other. 1 The 
angular separation between the two images is then 6 = X/a. 
In case the angular separation is that between two spectral lines, 
= di f . Hence for X in eq. 8-13 may be substituted a di', so 
that the resolving power is 

a di' 



But by eq. 8-12 this is equal to a D, so that 

(8-14) 

provided the prism is set for minimum deviation. 

Thus the resolving power of a prism at minimum deviation 
depends on the character of the glass, the wave-length, and the 
effective thickness of the prism. Since the effective thickness 
depends on the refracting angle A and the aperture a, any increase 
in either will result in an increase in the resolving power. The 
limit to the value of A for a single prism is imposed by the 
necessity for keeping ;*' less than the critical angle of refraction. 
Sometimes the light is passed through two or more prisms in 
succession in order to obtain greater dispersion, but this involves 
other optical problems which limit the usefulness of the method. 

Further consideration of eqs. 8-12 and 8-14, both of which, 
it must be remembered, apply only in the case of minimum 
deviation, is desirable to point out that while the width a of the 
beam of light intercepted by the prism appears explicitly in the 
expression for D, it does not appear in that for R. Nevertheless, 
while the resolving power depends on the aperture in the case of 
the prism, the dispersion does not. This apparent paradox is 
because, in eq. 8-12, t is always proportional to a; if a is decreased 
in a certain ratio, the effective thickness t is reduced in the same 
ratio, and the dispersion will be unchanged. On the other hand, 
if a is reduced, R will be reduced in proportion, since R is equal 
to a D. 

The resolving power of a prism is not necessarily that of the 
spectrometer on which it is mounted. If the aperture of the 
system is limited by the sizes of the collimator and telescope, 

1 See Fig. 12-19o. 



94 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VIII 



or camera objective, the value of the resolving power will be 
smaller than that of the prism alone. In visual observations the 
quality of the vision, the accuracy of focal adjustments, and 
the judgment of the observer also enter to increase or decrease the 
numerical value of R. Also, the definition of limit of resolution 
given in the preceding paragraph, originally due to Rayleigh, 
is an arbitrary one, and does not agree with experiment in case 
the two lines under observation are unsymmetrical or quite 
different in intensity. The numerical value obtained by eq. 8-14 
should be taken as indicating only the order of magnitude of the 
resolving power of a prism. In particular cases the value 
obtained by actual observation of two lines which are just dis- 
tinguishable as separate may be larger than that calculated from 
Rayleigh 's criterion. 





(b) 



FIG. 8-3. -Two forms of the constant-deviation prism. 

4. The Constant-deviation Prism. In the measurement of 
spectra with an ordinary prism spectrometer used visually it is 
necessary to calibrate the instrument for a given setting of the 
prism, in order to obtain any degree of accuracy. Not only is 
this calibration time consuming, but the translation of the settings 
of the telescope into wave-lengths is a tedious process. To avoid 
these calculations, a constant-deviation spectrometer may be 
used. In this instrument the prism is constructed as shown in 
Fig. 8-3. For any angle of incidence i the light of the wave- 
length which is at minimum deviation emerges at right angles to 
the direction of the incident beam, after one total reflection in the 
prism. The prism may be made of two 30- and one 45-deg. 
prisms cemented together as shown by the dotted lines, or it 
can be cut from a single block of glass. A convenient construc- 
tion is to make it of two prisms as shown in Fig. 8-36. The 
advantage here is that the glass path is diminished and loss of 



SBC. 8-5] THE PRISM AND PRISM INSTRUMENTS 



95 



light by absorption reduced. This is particularly desirable when 
the prism is of quartz for use in the ultraviolet region of the 
spectrum. This type of spectrometer is often made with exceed- 
ingly precise adjustments for focusing and setting on particular 
wave-lengths; it is illustrated in Fig. 8-4. This is really a 
spectrograph, made for photographic purposes, but it is equipped 
with special eyepieces which can be substituted for the plate- 
holder. The prism is rotatable about a vertical axis so that any 
given wave-length region may be brought into coincidence with 
the cross hairs, or set at a particular point on the photographic 




EJCIT5IJT 



FIG. 8-4. Diagram showing optical path in monochromator. (Courtesy of 

Gaertner Scientific Co.) 

plate. This rotation is controlled by a micrometer and screw, 
accurately calibrated in angstrom units. Since particular wave- 
lengths may be brought precisely to a given point in the field, 
this instrument is also called a monochromator, since a suitable 
aperture may be placed in the focal plane of the spectrum which 
isolates in turn those wave-lengths which fall upon it as the 
prism is rotated. 

6. The Direct-vision Spectroscope. Because of the primary 
importance of accuracy of measurement, ordinary spectroscopes, 
being massive and rigid, are heavy in construction and unwieldy 
in shape. For work which does not require a high degree of 
precision, a lighter and less cumbersome instrument, the direct- 



)6 LIGHT: PRINCIPLED AND EXPERIMENTS [CHAP. VIII 

vision spectroscope, is used. If two prisms, one of flint glass and 
,he other of crown are placed together as shown in Fig. 8-5, the 
lifference of dispersion will result in a difference of deviation and 
;he production of a spectrum, for if the deviations of the two 
jrisms are equal for one wave-length, they will not be HO for any 
)ther wave-length. To obtain sufficient dispersion it is custom- 
iry to cement together three or five prisms, as illustrated in 
ftg. 8-6. It is evident that the entire optical system is nearly 
n a straight line. The composite prism is usually cemented, 
>r otherwise firmly fixed, in a tube provided with a slit at one end, 
in appropriate collimator and telescope, and cross hairs or scale 
or calibration. Since the whole instrument can be pointed 
;asily at a source and used in a manner similar to a monocular, it 
s extremely useful for rapid visual inspection of spectra. Some 
>f the more elaborate direct-vision spectroscopes may even be 
ised for measurement of spectral-line positions with an accuracy 
)f about 1 angstrom. 





FIG. 8-5. FIG. 8-6. The direct-vision prism system. 

6. Critical Angle of Refraction. If the angle of incidence of a 
Deam of light on a glass surface is increased until it approaches 
)0 deg., the angle of refraction will approach a limiting value 
ivhich depends on the indices of refraction of the glass and of the 
ur traversed by the incident beam. In the case of a glass plate 
n air, the index of the air may be taken as unity, so that for 
> = 90, the index of refraction of the glass is given by 

n = _^L_, (8-15) 

sin r c 

where r c is called the critical angle of refraction. For glass whose 
index is 1.5, r c is approximately 41.8 deg. Consequently any 
light incident from the glass side on the air-glass interface at an 
angle greater than r c will be totally reflected, since there can be 
none refracted. Total-reflection prisms so constructed as to take 
advantage of this principle are often better than mirrors for 
turning beams of light through right angles. The reflecting 
power of most metal mirrors is far from unity and varies with 



SBC. 8-7] THE PRISM AND PRISM INSTRUMENTS 



97 



the wave-length. The total-reflection prism is free from dis- 
persion, since the angle of reflection is independent of the wave- 
length. On the other hand, a glass prism absorbs some light, 
and for regions of the spectrum not transmitted by ordinary 
glass the prism must be made of quartz or fluorite. In some 
cases rock salt or lithium fluoride prisms are used. 

It is necessary that the reflecting surface of the prism be free 
from dirt, oxidation, or other contamination, since the presence 
of a film other than air will change the critical angle, and more 
often than not cause light to be refracted out of the prism. By 
means of prisms of special design, the light can be turned through 
angles other than a right angle. 1 




FIG. 8-7. Critical angle of refraction. If the direction of the light is reversed, all 
rays incident on AB at angles greater than that for ray a will be totally reflected. 

7. Index of Refraction by Means of Total Reflection. The 

phenomenon of total reflection of light provides a useful means 
of determining the index of refraction of transparent substances. 
If a prism is illuminated by a broad beam of convergent light 
as shown in Fig. 8-7, the field at E will be divided into a dark 
portion on one side of the ray a and a bright portion on the other. 
If the refracting angle A of the prism is measured, and the angle 
i' between the normal to the surface AC and the emergent ray a, 
and these two values substituted in eq. 8-20, the index of the 
prism may be calculated. It will then be possible to measure 
the index of refraction of a liquid placed in contact with the 
side AB. 

Let the index of refraction of this medium be n, and that of the 
glass prism, n a . Then Snell's law, 

1 A fairly complete discussion of total-reflection prisms of a variety of 
designs is found in Bureau of Standards Scientific Paper No. 550, 1927, by I. C. 
Gardner. 



98 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. VIII 

n sin i = n g sin r, 
at grazing incidence becomes 

^ = -4 (8-16) 

n sin r 

At the surface AC the prism is in contact with air, and SnelFs 
law for this case is 

sin i' = n g sin r'. (8-17) 

Also 

A = r + r'. (8-18) 

Eliminating r and r' from these three equations, we have 

n = sin A\/n a 2 sin 2 i' sin i' cos .A. (8-19) 

The measured values of A, i', and n g substituted in eq. 8-19 will 




FIG. 8-8. 

give the value of n, the desired index of refraction of the liquid. 
The value of n , the index of refraction of the glass prism, may 
be found with great precision for any wave-length by measuring 
the angle A and the minimum deviation A, and substituting their 
values in eq. 8-5. This method is to be preferred to the measure- 
ment of n g by a measurement of i' for grazing incidence as sug- 
gested above. If, however, the latter method is to be used, 
eq. 8-19 may be put into a suitable form by putting n = 1, 
whereupon 

[ ^ ~1 2 

sin i + cos A 
sin A J 



SEC. 8-8] THE PRISM AND PRISM INSTRUMENTS 99 

8. The Abbe Refractometer. This is an instrument designed 
to make use of the principles outlined in the two preceding 
sections for measuring the indices of refraction of liquids. While 
many refinements are built into the best instruments, the essen- 
tial optical part is a pair of right-angle prisms illustrated in 
Fig. 8-8. When placed with their diagonal sides face to face 
with a thin film of liquid between, the index of refraction of the 
liquid may be found by measuring the angle i 9 corresponding to 
the light which enters the prism A and is refracted at the critical 
angle. Substituting of this value, and the measured values 
of A and n g in eq. 8-19, yields the value of the index of refraction 
of the liquid. * 

Problems 

1. Show that the constants of the Cauchy dispersion formula are 
given by 



/&1A1 J W 2 X 2 2 
A = ^1 2 ~^~X7~ ' 

J5 = r - c-~ 

AI AI> 

in which HI and n 2 are the indices of refraction at two wave-lengths Xi 
and X 2 . 

2. What will be the dispersion of a 60-deg. prism made of glass No. 3 
in Table 2 at the end of this volume, at 7000 angstroms? At 4000 
angstroms? Give the units in each case. If the prism face is com- 
pletely filled with light, how wide must it be if the sodium doublet 
5890 and 5896 angstroms is to be just resolved? 

3. What factors actually enter into an experimental determina- 
tion of resolving power other than those considered in the preceding 
problem? 

4. A prism for a spectrograph is to be made out of glass whose index 
UD is 1.72. What is the maximum prism angle which can be used? 

5. The critical angle of refraction of a substance is 58 deg. What is 
its index of refraction? 

6. Show that if the angle of a glass prism is larger than twice the 
critical angle of refraction, no light can be passed through it by refraction. 



CHAPTER IX 
THE NATURE OF LIGHT 

1. Light as a Wave Motion. Speculations, theories, and 
investigations concerning the nature of light have had a promi- 
nent place in man's intellectual endeavors since the beginning of 
history. There havejboeh short periods of time when groups, 
sometimes including practically all students of natural philosophy, 
have been convinced that-the nature of light was understood. 
On the whole, however, during most of the time, diverse opinions 
have been held, based on conflicting theories and speculations or 
on apparently conflicting experimental evidence. It is not 
within the province of an intermediate course in optics to present 
the history, or the arguments concerning different theories, of 
the nature of light, or, more generally, of radiation. But just 
as in the introduction to geometrical optics the concept of the ray 
was adopted because it enabled us to continue expeditiously our 
development of the subject of image formation, so in physical 
optics we can adopt the concept of light as a wave motion propa- 
gated from a source in all directions through space. Moreover, 
we can make use of the wave theory only as long as it is not in 
conflict with observations, whether these be in the limited field 
of the topic under discussion or in some other part of the larger 
field of physical phenomena. 

The quantum theory introduces a concept of light which is 
more complex than a mere wave motion. According to this 
view, when light is emitted or absorbed, the energy of the 1 
light appears in the form of concentrated units, called photon^. 1 
These photons are supposed to move in straight lines, when in 
free space, with the speed of light, and to have an energy which 
is related in a simple manner to the frequency of the associated 

1 It is perhaps worthwhile to warn the student against the indiscriminate 
use of the words " photon " and "quantum. 1 ' A photon consists of a certain 
amount or quantum of energy,*? but not all quanta are photons. 

100 * 



SBC. 9-l| THE NATURE OF LIGHT 101 

light wave. In problems of the transmission of light, where no 
interchange of energy between radiation and matter is involved, 
the quantum theory, like the classical wave theory, describes the 
propagation of the light in terms of a wave motion. It is prob- 
lems of this kind, including mainly refraction, diffraction, inter- 
ference, and polarization, with which we shall be concerned in the 
next few chapters. For these purposes, therefore, the assump- 
tion of light waves is entirely adequate, and it is not necessary 
that we concern ourselves with the complementary assumption 
of the existence of photons. On the other hand, the origin of 
spectra, the interaction of light with material media through 
which it passes, and certain phenomena classified under the 
headings of magneto- and electro-optics cannot be satisfactorily 
explained by the classical wave theory of radiation. For these, 
the quantum theory signalized by the names of Planck, Ein- 
stein, Bohr, and others offers a satisfactory explanation. This 
early quantum theory, however, in turn fails to encompass all 
the intricacy of detail in modern observations in the field of light. 
To take its place has arisen what is known as quantum mechanics. 
While this later quantum theory goes far in unifying the classical 
and earlier quantum concepts, we have not lived long enough 
with it to reduce it to simple terms. Accordingly, for an ele- 
mentary presentation, it is necessary to rely upon classical or 
quantum theories in turn to "explain" those phenomena to 
which they are individually best fitted. This process is, however, 
not entirely without a satisfactory basis, for, it will be noted, in 
order to deal with either the origin of the radiant energy (as 
photons) in atoms or molecules, as in spectra, or its interaction 
with material media, as in the photoelectric effect, the quantum 
theory is more suitable, while the classical wave theory is quite 
sufficient to explain those light phenomena which deal only with 
the passage of the light through space. In diffraction, where we are 
accustomed to thinking of a material obstacle as taking part in 
what happens, it is entirely immaterial of what elements the 
obstacle is made; the important detail is that a part of the 
"front" of the light propagated through space is obstructed and 
cannot pass on to the place where the image is formed. Even in 
polarization, where the nature of the medium assuredly enters 
into the entire problem, we can describe the characteristics of the 
transmitted light adequately by means of the classical theory. 



102 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. IX 

Experimental evidence supports the hypothesis that light is 
a wave motion, transverse in character, propagated through 
space with a finite velocity. The theory of light as a transverse 
wave motion has gone through several phases. In the earliest 
of these, it was felt absolutely necessary to suppose space to 
consist of an elastic-solid medium of great rarity, patterned in its 
characteristics after those substances which were known to be 
the medium of transfer of other disturbances such as sound and 
water waves. This elastic-solid medium could, therefore, be 
considered to consist of particles obeying the same laws as ordi- 
nary matter, but in such a way that suitable density and elasticity 
could be assigned to the medium. Later, this hypothetical 
medium was abandoned in favor of the more abstract idea of an 
all-pervading "luminiferous ether," different from the elastic- 
solid medium in that it possessed no such definite " particle'' 
characteristics, but retaining the properties of elasticity and 
density so necessary for the representation of the wave motion 
which traverses it. With Maxwell's introduction of the elec- 
tromagnetic theory of light, this elastic medium was replaced by 
one with the electrical characteristics of a dielectric constant 
and a magnetic permeability. On this view the wave has an 
electric field and a magnetic field, each transverse to the direc- 
tion of propagation and perpendicular to the other. This elec- 
tromagnetic theory of light waves has now completely superseded 
the idea of waves in an elastic ether. The ether, if we continue 
to use the term, is now thought of as a region with certain elec- 
trical characteristics rather than as an elastic solid. In this sense 
the idea of transverse waves in the ether is in accord with the 
latest developments of relativity and quantum theory. 

2. Velocity, Frequency, and Wave-length. The velocity of 
light in free space is the same for all wave-lengths. This con- 
clusion is supported by a variety of observation. A wave dis- 
turbance propagated through space with a velocity c and a 
frequency n will have a wave-length X. These three quantities 
are related by the equation 

c = nX. (9-1) 

The value of c is approximately 3 X 10 10 cm. sec." 1 ; hence for a 
wave-length X = 5 X 10~ 6 cm., the frequency n will be 0.6 X 10 15 
sec." 1 . If the time it takes a point on the wave train to pass 



SEC. 9-3] 



THE NATURE OF LIGHT 



103 



through a complete cycle of phases is called the period T, then 
n = 1/T, and 



_ X 
C ~ T' 



(9-2) 



3. Simple Harmonic Motion. It has been pointed out in 
Sec. 9-1 that our concept of the nature of a light disturbance has 
passed beyond the stage at which it was considered to be an 
oscillatory displacement of material particles. The form of 
analytical expression, however, need not be changed. At a point 
in space, the disturbance due to the passage of a train of light 
waves may be a simple or a complex wave motion. Also, it can 
be shown that a complex oscillatory motion may be represented 




FIG. 9-1. Illustrating simple harmonic motion. 

as a summation of a number of simple harmonic motions. We 
may therefore arrive at equations describing wave motions by 
the development of the summation of a number of simple har- 
monic disturbances of a material particle. 

The equation for a simple harmonic motion may be obtained 
by considering the motion executed by a point P moving with 
uniform angular speed in a fixed circle. The projection of this 
motion upon a diameter of the circle is a simple harmonic motion. 
We may thus consider the motion of a particle S in a straight line 
(Fig. 9-la) to be a simple harmonic motion, provided the dis- 
placement of S is always given by 

s = a sin w<, 

in which a is the maximum displacement, i.e., the radius of the 
circle, co is the angular velocity of the point P, and t is the time 
which has elapsed since the particle left the point in its upward 
journey. If T is the time taken for one complete cycle, then 



104 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. IX 

w = 27T/7 7 , and the displacement will be 

s a sin 2*7=- (9-3) 

In Fig. 9-16 the displacement is plotted as a function of the time, 
the solid curve being the graph of eq. 9-3. In general, however, 
it is desirable to express the displacement in terms of the time 
which has elapsed since the beginning of the motion, i.e., since the 
particle was at some other point such as S'. In eq. 9-3, however, 
it is assumed that the particle is at the position of zero displacer 
ment at the beginning of the time t. In general this will not be 
true. To represent the most general case, therefore, we must 
consider that the particle is at some position S' when t = 0; 
the displacement s is, as before, the distance from to S, and 
instead of eq. 9-3, we have 




s = a sin (POP' - BOP") = a sin I 2r - <p I (9-4) 



If the point S' should be above the middle point 0, as S", the 
value of <p is positive. The dotted curve in Fig. 9-16 is the graph 
of eq. 9-4. 

4. Phase and Phase Angle. The phase of a simple harmonic 
motion refers to the particular stage of the cycle of motion being 
executed. Two particles executing simple harmonic motions 
parallel to AA' in Fig. 9-la are in the same phase if they are at 
the positions of zero displacement at the same time and are also 
moving in the same direction; if moving in opposite directiorife 
they are in opposite phase. It is not necessary, however, that 
the simple harmonic motions be executed in parallel lines nor 
that they be of the same amplitude. They must be going through 
the same part of their cyclical motion at the same time, so that 
the equations for the displacements are the same functions (sines 
or cosines) of the same angles. The motions represented by eqs. 
9-3 and 9-4 are not in the same phase, the difference of phase 
angle between them being <p. On the other hand, the motions 
given by 



x = a sin 



(34 - )' 

y = 6 sin (2^ - <f), 



SEC. 9-5] 



THE NATURE OF LIGHT 



105 



are in the same phase, even though they are along x- and y-direc- 
tions perpendicular to each other, and have different maximum 
amplitudes. 

Obviously the phase angle corresponding to the displacement 
passes through all values from to 2ir in succession, repeating this 
change as long as the motion continues. It follows that the 
phase angle in eq. 9-4 is given by 



5. Composition of Simple Harmonic Motions. There are two 
cases to be considered: The composition of (1) simple harmonic 




(a) (b) 

FIG. 9-2. (a) Graphical method of composition of two simple harmonic 

motions. (6) The solid line gives the resultant of two simple harmonic 
motions which are shown by the dotted lines. 

motions in the same direction, and (2) simple harmonic motions 
at right angles. All cases come under these two heads, since 
two or more motions at an angle not nor 90 deg. can be sepa- 
rately resolved into components at right angles, which may 
thereupon be composed. There are two general methods, the 
graphical and the analytical, for effecting this composition. 
The graphical method will be discussed first. 

Graphical Methods. The composition of two simple harmonic 
motions of the same period T, executed in the same direction, 
but not necessarily of the same amplitude, may be represented as 
in Fig. 9-2a. The displacements Si and s 2 differ in phase angle 

by P 2 OPi. The total displacement OS along A A' is given by the 
projection of the resultant radius OR, which is the diagonal of the 



106 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. IX 



parallelogram formed by the radii OP\ and OP 2 . Also, 
OS = OS i + OSz- The individual simple harmonic motions 

and their resultant are plotted in 
Fig. 9-25. 

While this method may be applied 
successively to as many components 
as desired, a much easier method is to 
make use of the vector polygon, illus- 
trated in Fig. 9-3. The lengths of the 
vectors are the amplitudes of the 
separate components; the angles a\ t 
2 , etc., are the phase angles; OR is 
the amplitude of the resultant, and 

FIG. 9-3. Vector addition of ^ 

simple harmonic motions in its phase angle is ROB. This method, 
8ame line ' which may be extended to give the 

summation of any number of components, is extremely useful in 
the solution of problems in diffraction and will be made use of in 
Chap. XII. 





FIG. 9-4. Composition of two simple harmonic motions at right angles. 

If the two component vibrations are at right angles, they may 
be compounded graphically as illustrated in Fig. 9-4. The 
basis of the method is to make use of a series of equidistant points 
on circles drawn concentrically, with their radii equal to the 
maximum amplitudes of the two disturbances. These points are 
numbered so that the zero point in each case corresponds to the 
displacement for which t = 0. The values of the simultaneous 



SEC. 9-5] THE NATURE OF LIGHT 107 

displacements corresponding to the number k may be called 
x k and yk, respectively. In order to get the displacement due to 
the composition of the two disturbances at any instant i, it is 
necessary to find on this coordinate diagram the position of the 
point (xi,yi). The resultant disturbance will be the curve 
plotted through a series of points thus found. 

Analytical Methods. As in the case of graphical methods there 
are two general cases to be considered: (a) when the vibrations 
take place in the same direction, (b) when at right angles. The 
first of these is important in diffraction and interference, which 
are dealt with in the chapters immediately following this. Since 
the only problem of vibrations at right angles with which we 
have to deal is in the case of double refraction in crystals, case 
(6) will not be discussed here. The special case referred to will 
be found in the treatment of elliptically polarized light, Sec. 
13-11. 

For case (a) consider two simple harmonic motions executed 
in the same direction with the same period T. They may be 
represented by 



(9-5) 




The difference in phase between them is given by <p\ 
Expanding each sine term in eq. 9-5, and adding, 



s = Si -h 2 = ai(sin 6 cos <pi cos sin < 

4- 2(sin 6 cos v>2 cos 6 sin v> 2 ), (9-6) 

in which for convenience the symbol has been substituted for 
2irt/T. At this point it is convenient to choose an angle 8 such 
that 



A cos 5 = ai cos <p\ + a 2 cos , ,q -, 

A sin 6 = ai sin ^i + a sin 



If the first of eqs. 9-7 is multiplied by sin 6 and the second by 
cos 9, and the second subtracted from the first, 



A (cos 5 sin sin 6 cos 6) a\ sin cos <pi -f a 2 sin 6 cos <p 

ai cos 6 sin <f>\ a 2 cos 6 sin 



108 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. IX 

in which the right-hand side is the same as the right-hand side of 
eq. 9-6. Hence, 

s = s\ -f 2 = Al sin 2ir-=j cos 5 cos 2?r~ sin 8 J 

= A sin \2irt - A (9-8) 

Also, if eqs. 9-7 are squared and added, the result is 

cos (<p* - >i). (9-9) 



It is evident that A is the resultant amplitude of the composition 
of the two simple harmonic motions, for if <? 2 <f>\ is zero or any 
integral multiple of 2ir, the disturbances are in the same phase 
and A 2 = (a\ + a 2 ) 2 ; while if w <pi is equal to (2n l)/2 
times 2rr, where n is a whole number, the disturbances are in the 
opposite phase, and A 2 (a\ a 2 ) 2 . In this case, the amplitude 
will be zero provided a t = a 2 . 

6. Characteristics of a Wave Motion. Although, as has been 
pointed out in Sec. 9-1, light waves are no longer thought of as 
disturbances in an elastic-solid medium, in order to develop 
the equation of a wave motion adequate for present purposes, 
we may consider the form of the wave to depend upon the motion 
transmitted to the particles of such a medium. A particle 
moving with a harmonic motion of the kind described in the 
previous sections will act as the source of a wave train. Let O 
(Fig. 9-5) be a source communicating its harmonic motion to a 
medium having an elasticity E and a density d. The velocity 

of propagation of the wave is e = \/E/d. The displacement at 
a given instant along the line OS is given by s = a sin 2trt/T, 
and the displacement at the same instant of a particle from a 

(t t'\ 
point X a distance x from is given by 6 = a sin 2r( -~, ~~ ~f)' 

where t' is the time it takes the wave to travel from to X. In 
other words, the difference of phase between the motions at 
and X is given by t' IT. But since the time taken by the wave 
to travel from to X is t' = x/c, and since c = \/T, t'/T = x/\. 
Hence the displacement at X is given by 



W' *} 

H* ~ x| 



a sin ZrU; - r (9-10) 



SBC. 9-7] 



THE NATURE OF LIGHT 



109 



Although eq. 9-4 bears a superficial resemblance to eq. 9-10, 
it is not the same. The former gives the displacement at any 
time t of a single vibrating particle; the latter gives at any one 




FIG. 9-5. Illustrating the characteristics of a simple wave motion. 

instant an instantaneous "snapshot" of the displacements of all 
the particles along the path of the wave. 

The difference of phase between the motions at and X is 
x/\' } if this quantity is a whole number, the motions at the two 
points are in the same phase. Two particles at points X\ and 
X 2 will execute motions whose difference of phase is given by the 
difference of their phase angles 



(9-11) 




Distance 
Fio. 9-6. Superposition of two wave trains traveling in the same direction 

7. The Principle of Superposition. If light from two sources 
passes through a small opening at the same time, two separate 
images will be formed, each of which will in no way be affected 
by the presence or absence of the other. This will be true, 
unless the sources are so close together that their images overlap, 
even though at the opening the wave trains pass through the 



110 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. IX 



same space at the same time. Hence we may say that if two 
or more wave trains travel through the same space at the same 
time, each will thereafter be the same as if the others were not 
present. At a point where they act simultaneously, howwver, 
the resulting disturbance will be that due to the superposition 
of the wave trains. This is illustrated graphically in Fig. 9-6, 
where the dotted curves represent the separate disturbances and 
the solid curve the result of their superposition at any one 
instant. 

8. The Wave Front. A simple concept of a wave front is 
that of a surface A (Fig. 9-7), traveling from a source S. From 



Source 




FIG. 9-7. Illustrating the Huygens principle. 

S a disturbance which is now at A spreads out through the 
medium. Subsequent vibrations at S set up succeeding wave 
fronts. With this view, the wave front may have any shape 
whatever. It follows, too, that if the source is sending out 
oscillatory disturbances to all parts of the wave front, the 
motions at all points in it will be in the same phase. It is not 
necessary that the medium be homogeneous ; the wave front may 
lie part in one medium and part in another. Rigorously, how- 
ever, the definition breaks down if different wave-lengths are 
propagated in any of the media with different speeds. In such 
a case we may continue to use the term wave front only with 
regard to homogeneous waves. 

9. The Huygens Principle. Secondary Waves. In order to 
account for the manner in which light waves are propagated, 



SBC. 9-11] 



THE NATURE OF LIGHT 



113 



energy upon a plane normal to the direction of propagation is 
proportional to the square of the amplitude of the disturbance, 
and is also defined as the intensity of the light in that plane, it 
follows that the intensity is proportional to the square of the 
amplitude. Figure 9-9 shows graphically the relation between 
the two. The dotted curve is the graph of S = cos x, the maxi- 
mum amplitude being unity. The solid curve is the graph of the 
intensity / = S 2 cos 2 x. 




FIG. 9-9. Amplitude in a wave train is indicated by the dotted curve, intensity 

by the solid curve. 

11. The Velocity of Light. The first determination of a finite 
velocity of light was made by Romer, who in 1676 noted that 
inequalities in the time intervals between eclipses of Jupiter's 
satellites depended upon whether the earth was on the same side 
of the sun as Jupiter, or on the opposite side. In the former case 
the eclipses occurred earlier, and in the latter case, later than the 
predicted times. Romer inferred that the difference was because 
the time taken by the light from Jupiter to reach the earth is 
finite, and greater when the two planets are farther apart. His 
calculated velocity was a little over 300,000 km. per sec. His 
conclusion was ignored by many until in 1728 Bradley discovered 
the so-called aberration of light. This is really an aberration in 
the positions of fixed stars, which were found to have slight dis- 
placements in position, depending on the motion of the earth in 
its orbit. The effect is illustrated in Fig. 9-10. When the earth 
is moving to the left, in order to bring the star image on the 
center of the field, the telescope must be pointed a little forward 
in the direction of the earth's motion, that is it must be pointed 
a little to the left in the figure, while at position B, it must be 



114 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. IX 



pointed a little to the right. The angle of aberration a. is about 
20.5 sec. of arc. Bradley concluded that this alteration of the 
apparent direction of the light was due to the relative velocities, 
c of the light and v of the earth. The relation between these 
velocities and a is given by tan a = v/c. This gives a value of a 
little less than 300,000 km. per sec. for c. 

In 1849, Fizeau made a preliminary experimental determina- 
tion of the velocity of light, using the toothed-wheel method, 
illustrated schematically in Fig. 9-11. A simple illustration will 




Earth's 
orbit 



v \ 




FIG. 9-10. Illustrating how the angle of aberration arises. During the time 
the light traverses the length of the telescope, the latter moves from the position 
indicated by the dotted outline to that given by the solid outline. 

suffice to show the manner in which this device may be used to 
measure c. Suppose the light beam passes through a slot on 
the rim, is reflected from the distant mirror, and returns on the 
same path just as the next slot is exactly in position to receive it. 
Then the ratio of the velocity of the wheel's rim to the velocity 
of light is the same as the ratio of the distance between the two 
slots to the distance traveled by the light, which is 2TM in the 
figure. 

In the year following Fizeau's experiment, the rotating-mirror 
method was used by Foucault. This method is shown schematic- 
ally in Fig. 9-12. Light from the source S is reflected from the 
rotating mirror R to a distant mirror M. The center of curvature 



SEC. 9-11] 



THE NATURE OF LIGHT 



115 



of M is at R so that with a stationary mirror R the light will 
be reflected directly back on its path, to S. Actually it returns 
to /Si, having been reflected by mirror Af 2 . If R is turning at 
high speed, in the time during which the light passes from R to 
M and back, the mirror has turned through a small angle, so that 





FIG. 9-11. Fizcau'e toothed-wheel apparatus. T is the toothed wheel; Li 
collimates the beam of light; Li at the distant station focuses the light upon the 
mirror M; the source is at <S; and the eyepiece or telescope at E. 

the return beam, instead of arriving at Si, is observed at Si, a 
small distance from Si. From a measurement of S\Si, the 
angle through which R has turned may be calculated. If also 
the angular velocity of R and the distance 2RM are known, the 
velocity of light may be found. 




FIG. 9-12. Foucault's rotating-mirror apparatus. 

The actual experimental technique and calculations involved 
in both Fizeau's and Foucault's experiments are far more than 
the bare details just given, and the reader is referred to more 
extended treatises for their complete description. 1 

1 See, for instance, Preston, "Theory of Light," 4th ed., Macmillan. 



1 16 LIGHT: PRINCIPLES AND EXPERIMENTS (CHAP. IX 

The experiments of Fizeau and Foucault were preliminary 
trials of their respective methods. In the years following, both 
methods were used extensively to find the velocity of light. l The 
method of Foucault has proved to have experimentally fewer 
inherent objections. Of these, two are worthy of mention. 
When the lens L is placed as shown in Fig. 9-12, the amount of 
light returned to S' is inversely proportional to RM, since in any 
given revolution of R the light sweeps around on a circumference 
4irRM . In order to avoid this difficulty, Michelson moved the 
lens to a position between R and M and close to R. Another 
difficulty of the Foucault method is the possibility of error in 
i|jfi|wuring the very small displacement SS'. This and other 
optical difficulties were eliminated by Michelson in his final 






FIG. 9-13. Michelson's final apparatus for measuring the velocity of light. 
The path of the light beam is S, R, Mi, M, Ms, M*, M:,, M\, Ms M , Mi, R, 
P, T. 

series of experiments, carried out at Mt. Wilson during the past 
decade. The final form of his apparatus is shown in Fig. 9-13. 
In order to avoid excessively high speeds of rotation, mirrors of 
8, 12, and 16 faces were used. These were rotated at such speeds 
that while the light reflected from R was traveling over its journey 
to the distant station and back to R, the latter turned through an 
angle equal to that between two faces. Thus by a sort of "null- 
point" method the measurement of the image displacement SS f 
(Fig. 9-12) was eliminated. 

With this apparatus, and a light path between mountain peaks 
of about 35 km., Michelson obtained a value for c of 299,796 km. 
per sec. 2 This value is the velocity of light in vacuo and is 

1 There is a complete table of experimental values obtained by different 
workers, and references to original sources, in an article by Gheury de Bray, 
Nature, 120, 602, 1927. 

* Astrophysical Journal, 85, 1, 1927. 



SBC. 9-12] THE NATURE OF LIGHT 117 

obtained by adding to the observed velocity a correction for the 
index of refraction of the atmosphere. In a later experiment, 
carried to conclusion after his death in 1931, the light path was 
enclosed in an evacuated tube 1 mile long. By means of multiple 
reflections, the actual path was made eight to ten times as great. 
A rotating mirror with 32 faces was used. The mean value of 
many determinations was 299,774 km. per sec. 1 

These later determinations of c by Michelson and his associates 
were made with such a degree of precision that there was remark- 
able consistency between the individual observations of which 
the published values are the mean. Conservatively estimated, 
this consistency was between 50 and 500 times as great as in 
previous experimental determinations. 

The velocity of light has also been obtained by using the Ken 
cell (effect of electrical birefringence) as a shutter to cut off the 
light beam. By this method, which is discussed in Sec. 16-10, 
the value of c is found to be 299,778 km. per soc. 
' ^12. Wave Velocity and Group Velocity. Rayleigh was the 
first to point out that the velocity of light measured in a refracting 
medium is not the velocity of the individual waves. Instead, 
because of the difference of velocity with wave-length, the 
measured value will be that of a periodicity impressed upon the 
wave train. The velocity of this periodicity is called the group 
velocity. Consider a wave train having two wave-lengths, as 
illustrated in Fig. 9-14a, in which the dotted line represents the 
longer wave-length, traveling faster than the shorter. While at 
the instant represented the two are in phase at point A, giving 
rise to a group amplitude shown in Fig. 9-146, somewhat later 
the amplitude will build up a little to the left of the point A. 
In other words, the group will have a slightly smaller velocity 
than that of the individual waves. The energy belongs to the 
group rather than to the waves, and the observed velocity will be 
that of the group. 

The effect may be illustrated by the manner in which waves 
travel over the surface of water. It will be noticed that the 

1 MICHELSON, PEASE, and PEARSON, Astrophysical Journal, 82, 935, 1935. 
This series of experiments indicated also a monthly variation over a range of 
about 20 km. per sec., but the spread of the observations was sufficiently 
great to render its reality questionable. Whether this is real or is due to 
some instrumental effect is not at present known. 



118 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. IX 



group as a whole does not move as fast as the waves, which run 
forward and die out in the advancing front. 





(b) 



FIG. 9-14. (a) Two wave trains of different wave-length traveling toward the 

right; (6) the sum of their amplitudes. 

In order to obtain an analytical relationship between the 
wave velocity V and the group velocity U, consider two infinitely 



r/ x \ 

long trains of waves to be represented by sin 7=-! t =- ) 

^i\ KI/ 

2ir/ x \ 

[ t 77- ) Their resultant will be 

\ K 2 / 



and 



sn 



a . 2iri. x \ . . 2irf. x 

s = sm F,V ~ YJ + sm F,V ~ Y 

which can be put in the form 

S = cos 3r[i(l - 1) - l( f v i - ,^-J J X 



sn 



(9-13) 



If Ti is almost equal to T> 2) and V\ almost equal to 7 2 , the follow- 
ing approximations may be made: 



J_ 
T 2 



so that eq. 9-13 becomes 

e f*'^ 

S = cos TT - f^ 






. 2*1 
sin -=rl 



SBC. 9-12] THE NATURE OF LIGHT 119 

This is an equation in which the cosine part is a periodically 
varying amplitude factor, representing a wave group which 
moves with a velocity V equal to the ratio of dT/T z to 
d(TV)/T 2 V z , so that, using the relationship V = \/T, we obtain 

V*dT Vd\ - XrfF XdF 




~ d(TV) 

In an experimental determination of the velocity of light in 
carbon bisulphide, Michelson found that the ratio of the velocity 
in air to that in carbon bisulphide was 1.76. The ratio of the 
indices of refraction, however, gave a value of 1.64. The differ- 
ence is because the index of refraction is expressed as the ratio of 
the wave velocities, while, as pointed out above, the measured 
velocity is that of the group. By applying the correction term 
in eq. 9-14 the figures were found to agree. 

Problems 

1. Plot the graph of the simple harmonic motion given by 



s = 5 cos -p 

2. Using the parallelogram method, draw the graph of 

Si = ai sin f 2wjr and s 2 = #2 sin V^TT^T + 

representing two displacements along the same axis, for a\ = 5, a 2 = (3, 
T! = 2T 2 , <p = 7T/4. 

3. Draw a graph of the resultant motion of displacements 



x = a sin ( 27r 0i ), 

y = b sin 



where a = 7, 6 = 10, 0i = ?r/6, 6 2 = T/2. 



CHAPTER X 

INTERFERENCE OF LIGHT 

1. Interference and Diffraction Compared. In Sec. 9-9 it 
was shown that two Huygens wavelets will unite to produce light, 
provided the difference of path from their starting^ jjoint. to the 
point wherejjhey combine is equal to an even number of half 
wave-lengtfis ; and that they will unite to produce darkness if the 
difference of path is an odd number of half wave-lengths. Simi- 
larly it was shown that the effect of restricting t.hfi lighten a 
portion of the wave front by a narrow slit will produce an inten- 
srty~paElerh in Which the distribution of energy depends on the 
wave-length of the light and the width of the slit. All such 
patterns which similarly depend on the limitation of the wave front 
are called diffraction patterns. They may be shown to owe their 
appearance to the fact "that "in directions other than that of the 
incident wave there is not complete mutual cancellation of the 
light. A phenomenon bearing a superficial resemblance to that 
of diffraction is obtained if the beams of light from two separate 
parts of the wave front are made to reunite under conditions which 
will be described. The result is called interference of light. It is 
similar to diffraction in the sense that there exist alternate light 
and dark regions, depending on whether the two wave trains 
cancel each other, wholly or in part, or whether they reinforce 
each other. It is different from diffraction, however, since it does 
not necessarily depend upon any restriction of the wave front. 
Instead, the best interference patterns are produced with wave- 
fronts so extended that no diffraction phenomena of ordinary 
magnitudes exist. 

'2. Conditions for Interference. There are certain experi- 
mental conditions which must be fulfilled for the production of 
observable interference. These are: 

a. The light in the two wave fronts which combine to give interference 
must originally come from the same source. 

b. The difference of optical path between the beams must be very 
small, unless the light is monochromatic or nearly so. 

120 



SBC. 10-2] 



INTERFERENCE OF LIGHT 



121 



c. The wave fronts, on recombining to form interference patterns, 
must be at a small angle to each other. 

There is a fourth condition concerning the state of polarization 
of the light, which may be left to the chapter dealing with that 
subject. 

The first condition is made necessary by the nature of light 
itself. According to spectral theory, radiation of a particular 
frequency occurs when an atom or molecule undergoes a transi- 
tion from a given energy state to one of smaller energy. Such 







FIG. 10-1. 

transitions occupy a time of the order of 10~ 8 sec., during which a 
photon, or quantum, of radiant energy, passes out into space. 
The chance is believed to be extremely small that the same or 
another atom or molecule in another part of the source will emit 
a train of waves of any duration capable of producing interfer- 
ence with that from the first. 

The second condition may be illustrated by the diagrammatic 
representation in Fig. 10*1 of two interfering trains of white light. 
These are originally from the same source, but by some sort of 
apparatus the original wave train has been divided into two 
at a very small angle, advancing toward the right. The differ- 
ence of optical path from the source to the position A is zero, 



122 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. X 

and there the two waves, indicated by solid curves, will be in the 
same phase and will also reinforce each other throughout their 
paths. Since the difference of path from the source is zero, the 
reinforcement will take place for all wave-lengths. At a position 
B, where the difference of path is X/2, the two wave trains indi- 
cated by solid curves Are opposite in phase and will cancel each 
other, provided they are of the same amplitude, while some other 
pair of wave trains of different wave-length X 7 , indicated by the 
dotted curves, will not be opposite in phase and will to some 
degree reinforce each other. At a position C, as at A, there is 
again reinforcement for wave trains of wave-length X, and can- 
cellation for those of wave-length X 7 . Thus, for white light, only 
a few fringes are seen on either side of the middle position A. 
Outside of the region where fringes are seen, although at each 
point destructive interference exists for some wave-length, partial 
or complete reinforcement exists for all others. This results in 
a complete masking of all interference except for the few fringes 
already mentioned. Except for the middle one, these will be 
colored, with the dispersion increasing with increasing distance 
from the middle. 

If the source is one which contains only a few strong mono- 
chromatic 'radiations, interference fringes will appear with fair 
visibility over a considerable range on either side of the middle 
position. The mercury arc and the neon discharge tube are 
examples of this type of source. 

If there are only two wave-lengths, the fringes will have maxi- 
mum visibility at the point of zero difference of path, and also 
at the points where the path difference is an integral number of 
times one wave-length and also an integral number of times the 
other. At points in between, reinforcement will take place for 
one wave-length and partial or complete cancellation for the 
other, with the result that the visibility of the fringes will be 
low or zero. 1 There will be a more extended treatment of this 

1 The appearance or nonappearance of the fringes depends also upon the 
difference of color, i.e., upon the sensitivity of the eye for the two colors. 
If one wave-length is in the yellow, to which the eye is most sensitive, and 
the other in the deep blue, the fringes due to the yellow will be seen even if 
the intrinsic intensity of the blue is equal to that of the yellow. But by a 
proper adjustment of the intensity of each one, they may be made to cancel 
each other. 



SEC. 10-3] 



INTERFERENCE OF LIGHT 



123 



topic in Chap. XI, in the discussion of the Michelson 
interferometer. 

The third condition for interference, which applies rather to 
the observation than the production of the fringes, is illus- 
trated in Fig. 10-2ct and b. In a are represented two plane wave 
trains from the same source which have b&n made to cross each 
other at a small angle. At all positions indicated by solid lines, 
the phase is the same. It is different from this phase by a half 
period at all positions indicated by the dotted lines. Hence the 
crossing of two solid or two dotted lines marks a position of 





FIG. 10-2.- 



ii 



(a) (b) 

-Superposition of two plane waves (a) at a small angle, (6) at a larger 

angle. 



reinforcement, while the crossing of a solid and a dotted line 
marks a position of cancellation. In Fig. 10-26 the angle 
between the wave fronts is greater than in Fig. 10-2a, and the 
positions of reinforcement, or interference maxima of intensity, 
are closer together. As the angle between the two wave fronts 
is increased, the spacing of the fringes becomes smaller, until 
finally they are indistinguishable even with large magnification. 

3. No Destruction of Energy. The use of the term "destruc- 
tive interference" does not imply that where two wave trains 
from the same source cross each other some of the radiant energy 
is destroyed. At a position of minimum intensity, because of 
partial or complete cancellation of the amplitude, the intensity 
is very small, while at a maximum, since the amplitudes are 
added, the intensity, which is the square of the amplitude, is 



124 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. X 



correspondingly large. Conside^ a region of maxima and minima 
due to two wave fronts of eqifeil amplitude a, in which the inten- 
sity pattern is that shown graphically in Fig. 10-3. It is required 
to find the total illumination over a range from x = to x = k, 
as shown by the shaded area. According to eq. 9-9 this will be 
given by 



/ = f *"* (2a 2 + 2a 2 cos 

^/ X ** 



(10-1) 



where <p is the amount by which the difference of phase between 
the interfering wave fronts varies along the wave front between 

x and x = k. Since <p is a 
linear function of x, we may write 
<p = mx (see cq. 9-10). By inte- 
grating, we have 

. 2a 2 + sin mk 





m 



(10-2) 



2a' 

Intensify 



FIG. 10-3. 



Since the interference pattern is 
at a maximum at both and k, 
km = 2ir, the second term on 
the right is zero, and the total 
illumination in any such area 
of the field of view is / = 2a 2 k. 
This is exactly the illumination 
in the area if the two waves had combined without interference. 
Hence in the phenomenon of interference there is no destruction 
of energy, but only a redistribution. 

w/4. Methods for Producing Interference. Instruments for 
producing interference phenomena may be divided into two 
general classes: (A) Those which by reflection, refraction, or 
diffraction change the directions of two parts of the wave front 
so that afterward they reunite at a small angle; (B) those which, 
usually by a combination of reflection and refraction, divide the 
amplitude of a section of the wave front into two parts to be later 
reunited to produce interference. In both cases the usual con- 
ditions for interference must always be observed. In all instru- 
ments of class A it is necessary to use either a point source 
or a very narrow elongated source such as an illuminated slit 



SEC. 10-5] INTERFERENCE OF LIGHT 125 

parallel to the intersection of the two wave fronts. With instru- 
ments of class A, which may be^characterized as effecting a 
division of wave front, diffraction will also usually be observed, 
although often the spacing between the maxima and minima of 
the diffraction pattern is so large in comparison with the spacing 
between the interference fringes that it is easy to distinguish 
between the two effects. In the instruments of class B, which 
may be characterized as effecting a division of amplitude in a 
more extended portion of the wave front, it is not necessary to 
use a point or narrow line source. Since the wave front is 
divided in amplitude, if upon reunion corresponding points in the 
separate parts are superposed, the first condition for interference 
will hold. 

While the classification just given is probably the most impor- 
tant, all instruments for producing interfejncjg. patterns may 
also be grouped in two other categories, depending upon the 
existence or~nonexistence of a (^in^lernentaryjQattejn. In general, 
those of class ^n(3ivision of wave front) do notpossess comple- 
menlarjrpatterns, while those of classj? 



witEjt f ew^ceptioiis7 do^bssessjhem. Since a more extended 
disciissioiTof this distinction involves a description of the details 
of each instrument, it will not be carried out here. 

5. Young's Experiment. Historically the first true inter- 
ference effect to be recognized as such was due to Thomas 
Young. It belongs to class A, since the device he used recom- 
bined two different parts of the wave front so as to produce 
alternate light and dark fringes. His apparatus consisted of a 
pinhole to admit the light of the sun, and, in another screen a 
short distance away, two pinholes sufficiently close together so 
that the light diffracted at the first hole entered both of them. 
The arrangement is illustrated in Fig. 10-4. 

Diffraction also occurred at each of the two holes in the second 
screen, and in the overlapping portions of the diffracted wave 
fronts interference was observed. For best results it is more 
convenient to use narrow slits instead of pinholes, care being 
taken to make all the slits perpendicular to a common plane. 
The maxima and minima will then be evenly spaced bright and 
dark lines of equal width. The maxima will appear where the 
difference of path between the two wave fronts is an even number 
of half wave-lengths. Figure 12-136, page 177, in the chapter on 



126 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. X 



diffraction is a photograph of the pattern obtained with Young's 
apparatus, using slits instead of pinholes. The large scale 
pattern of maxima and minima is due to diffraction, while 
interference is responsible for the finer pattern which is most 
pronounced in the middle but extends over practically the entire 
field. 

The analysis of the conditions for the production of this inter- 
ference pattern will not be undertaken here. While the results 
are strictly due to what has been called interference of light, it is 
customary to treat the effects due to two or more parallel slits 
as extensions of the diffraction due to one slit. Since it is 
somewhat easier to handle the analysis in this way, a more com- 
plete discussion will be given in Chap. XII. 




'in 



Region of 
interference 



Fia. 10-4. Illustrating Young's apparatus. 

Among the devices for obtaining interference by a division 
of the wave front into two parts, three will be selected for dis- 
cussion, since they illustrate most generally the variety of condi- 
tions which may exist. These are the Fresnel mirrors, the 
Fresnel biprism, and the Rayleigh refractometer. 

6. The Fresnel Mirrors. The first of these is in a sense an 
adaptation of Young's apparatus, designed to eliminate as far as 
possible the presence of the diffraction pattern due to the narrow 
slits in the second screen. It is illustrated in Fig. 10-5. Light 
from a source 5, which is usually a narrow illuminated slit, 
passes to two mirrors which are inclined at a very small angle, 
with their line of intersection parallel to the slit S. The fringes 
may be seen by placing the eye near the mirrors so as to receive 
the reflected light, but ordinarily an eyepiece will be needed to 
magnify them. If monochromatic light is used, its wave length 
may be determined. Let a be the angle between the mirrors, 
Di the distance from the slit S to the intersection of the mirrors, 
Z> 2 the distance from the mirrors to the point of observation, and 
e the distance between two adjacent bright fringes F\ and F*. 



SBC. 10-6J INTERFERENCE OF LIGHT 127 

The difference of path between the distances SFi and SF 2 is 
X. The light appears to come from two virtual sources' &i and 
S z , whose distances from the point of observation are DI -+ D 2 , 
and whose linear separation may be called d. Since the, angle 




FIG. 10-5. Illustrating the Fresnel mirrors. 

between the reflected beams is twice that between the mirrors, to 
a sufficient degree of approximation, 

2 = (10-3) 

Also, 

3 " 5TTTS' (1 - 4) 

Combining these, 

2aeDi 



Since all the dimensions on the right-hand side of eq. 10-5 may 
be measured with considerable accuracy, a fairly precise value of 
the wave-length X may be found. Usually the mirrors are set so 
that the angle of incidence is large. In this case the angular aper- 
ture subtended at the source slit by each mirror is small, and 
diffraction is present. An added drawback is that the diffraction 
pattern has approximately the same spacing as the interference 
pattern, so that the two are not always distinguishable. A 
photograph of the fringes obtained with the Fresnel mirrors is 
shown in Fig. 10-6. 



128 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. X 




S 
1 



I 



SBC. 10-7] 



INTERFERENCE OF LIGHT 



129 



7. The Fresnel Biprism. A much better device for obtaining 
the interference between two sections of a wave front, with 
diffraction either largely eliminated or distinctly separated in 
appearance from the interference pattern, is the Fresnel biprism, 
illustrated in Fig. 10-7. The biprism is usually made of a single 
piece of glass so shaped that it is in reality two triangular prisms 
base to base, with equal and small refracting angles a. The 
biprism is set so that it is illuminated by light from a slit S. In 
order that the interference fringes may be distinct, the refracting 
edges of the two prisms should be parallel to each other, arid the 




u 



^J 
^1 



Fio. 10-7. The Fresnel biprism. 



intersection of the two inclined faces should be accurately parallel 
to the slit. 

To find the wave-length X of monochromatic light, it is neces- 
sary to know the distance d between the virtual images Si and <S 2 
from which the rays bent by refraction seem to come, the dis- 
tances DI and Z>2, and the separation e of two adjacent bright 
fringes in the field of view at the cross hairs. The value of d 
may be calculated if the index of refraction n of the biprism and 
its refracting angle a are known. Since the angles are small, we 
may consider the light to be passing through the prisms at minimum 
deviation, whence 



n . 



sn 



sn 



(10-6) 



where 5 is the angle of deviation. Equation 10-6 may be put 
in the form 



. a .a d . a . 8 
n sm jr ** sm -x cos 5 4- cos 75 sm =;' 

i i i ft ft 



130 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. X 

which, since the angles are small, may be reduced to 

= - (10-7) 



But from the figure, sin l d = d/4Di = 5/2, approximately, 
so that 

d = 2D ia (n - 1). (10-8) 

The value of d may also be obtained experimentally, making it 
unnecessary to know the index of refraction of the biprism. If a 
lens whose focal length is less than one fourth of the distance 
#1 + DZ is placed between the biprism and the eyepiece, two 
positions may be found at which real images of the slit, one 
formed through each prism, may be focused at the plane of the 
crosshairs, where the fringe width e has been observed. From 

Si 
S 

& 

FIG. 10-8. 

elementary geometrical optics it may be shown that if d\ (Fig. 
10-8) is the separation of the images for the first of these lens 

positions and e? 2 that for the second, then d = \/did z . The 
distances D\, D 2 , and e may be measured directly. 

Having found d, we may proceed to find X. In Fig. 10-7 
consider the paths S\Fi and SzFi to be such that a bright fringe 
is formed at FI. Then, if the adjacent bright fringe is at F 2 , 
the paths SiFz and SzF z will differ by the wave-length X. From 
the geometry of the figure, 




8m * = d = 
or 



This equation applies in the case where the light from the slit 
incident upon the prism is divergent. If the incident light is 
changed into a parallel beam by means of a collimating lens 



SBC. 10-7] 



INTERFERENCE OF LIGHT 



131 




132 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. X 



after leaving the slit, Di is very large compared to >z, so that the 
latter may be ignored, and eq. 10-9 becomes 

ed 



Substituting in this the value of d given in eq. 10-8, 



X = 



or, finally, 



X = 2ea(n - 1). 



(10-10) 



But for small prism angles a(n 1) = 6, the angle of deviation, 
so that 

X = 2e8. (10-11) 

The fringes obtained with a biprism are shown in Fig. 10-9. 

f. The Rayleigh Refractometer. This instrument, illus- 
trated in Fig. 10-10, has been used extensively for the determina- 



,* Tubes 


1 


f Oj 


\ 


1 






t 


~~" ~"~ *~~ *. XI 


E. 




~" ^ vr" 






- - """" "" -r 






E 


'' S' 




A. 




^ ^ 



Fio. 10-10. The Rayleigh refractometer. The figure in the circle illustrates 
the arrangement of the compensating mirrors; B\ and Bz are in front of the tubes, 
while Ba is below them. 

tion of indices of refraction of liquids and gases. While it 
possesses some of the general features of Young's apparatus, as 
do the Fresnel biprism and mirrors, there are important differ- 
ences to be noted. In the Rayleigh refractometer the two inter- 
fering beams are originally at a large angle when they leave the 
source Si, while in Young's apparatus they travel over nearly 
adjacent paths in a diffracted beam. In the second place, the 
two portions of the wave front passing through the slits <Sa and S* 



SEC. 10-8] INTERFERENCE OF LIGHT 133 

are collimated by a lens L\ so that they remain parallel for some 
distance. They then pass through a second lens Z/2, placed some 
distance from the first, which focuses the two beams, forming an 
image of the slit S\ at its principal focus. This image will, 
however, be a wide diffraction pattern similar to that shown in 
Fig. 12-14, page 179, with fine interference fringes superposed 
upon it. The greater the distance between Sz and $2', the finer 
will be the interference fringes. Since focusing in only one plane 
is required, the eyepiece E may be a cylinder with its axis parallel 
to the slits. Since this has a magnification in only one plane it 
gives a brighter image. The distance between LI and L 2 is 
made great enough, and the separation of Si and Sz is made wide 
enough, so that two tubes containing liquids or gases whose 
indices are to be compared may be placed side by side in the 
paths of the beams. The glass windows at the ends of these 
tubes must be of good optical glass with accurately plane faces. 

A change in the index of refraction of the substance in the 
tubes may be found as follows : Let us suppose that the two tubes 
contain a gas under the same conditions of pressure and tempera- 
ture. Given equal lengths, the optical paths are equal. A slight 
change in the conditions in one tube will cause a change in the 
optical path there, and hence a displacement of the fringes. A 
measurement of the amount of this shift, which under actual 
conditions is very small, is difficult and subject to uncertainty 
because of the narrowness of the fringes. Instead, it is customary 
to make use of a so-called coincidence method. At B\ and B 2 
in the paths of the beams are placed two plane- parallel plates 
of optical glass, of the same thickness, each at an angle of about 
45 deg. to the vertical. Under these conditions the optical paths 
through them are the same and will produce no displacement of 
the fringes. If, however, the index of refraction of the gas, and 
hence the optical path through B\ t is altered, the fringes, having 
been shifted on that account, may be brought back to their 
original position by a rotation of B\ about a horizontal axis 
perpendicular to the length of the tubes. Wheri this rotation is 
made, the optical path through B\ will be changed by an amount 
which is a measure of the index change. In order to provide a 
fiducial position to which the fringes may be brought back each 
time, there is placed across the lower part of the field, below the 
level of the tubes, a plate of glass B 3 , whose retardation is the 



134 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. X 



same as that of # 2 . Through this may be observed a set of 
fringes undisturbed by any changes in the tubes. The presence 
of this set of fringes as a fiducial system renders the use of a 
cylindrical eyepiece necessary, since one of the ordinary type will 
result in the superposition of all rays from a single source point, 
whether they pass through the tubes or below then % It is 
desirable to use white light, as the central bright fringe is easily 
identified, while over a considerable range all monochromatic 
fringes look alike. Moreover, because of selective absorption 
in the tubes, the fringes in the upper and lower parts of the field 
of view may not be the same in appearance. This makes it 
difficult to find accurately the fractional part of a fringe in white- 
light fringes, which are colored except for the central fringe. On 




(a) 




FIG. 10-11. 



this account it is customary to locate the central fringe of the 
system in white light, and substitute monochromatic light to 
measure the fractional part of a fringe. The motion of plate Bi 
must be controlled and calibrated with extreme accuracy by 
means of a micrometer screw. 1 

The Rayleigh refractometer has some serious drawbacks. 
It will be seen in the discussion of the Michelson interferometer 
that these are inherent in interference apparatus of class A, and 
are absent when those of class B arc used. On the other hand, 
the Rayleigh refractometer in its most modern form is still used 
a great deal for measurements of refractive, index, and at least 
one portable instrument is on the market. ' The principal draw- 
back is that the slits S 2 and >S 2 ' must be put as far apart as possible 
so that tubes of sufficient width may be used. It is also desirable 
that the tubes be sufficiently far from each other so that the 
physical conditions in them may be controlled separately. On 
the other hand, the farther apart the slits are, i.e., the greater 



1 In some forms of this instrument BI and Ba are fixed at right angles 
and turned together about a horizontal axis. 



SEC. 10-9] INTERFERENCE OF LIGHT 135 

the angle their separation subtends at the primary slit Si, the 
finer will be the fringes. The resulting loss of intensity when 
these are magnified makes measurements difficult. One of the 
most useful devices for increasing the separation of a pair of 
interfering beams without increasing the angle subtended at the 
primary slit is the biplate illustrated in Fig. 10-1 la. An ingenious 
application of the principle of the biplate has been made by W. E. 
Williams in a modification of the Rayleigh refractometer. l 

9. The Williams Refractometer. The essential feature of 
this instrument is illustrated in Fig. 10-116. Instead of passing 
through two narrow slits, each of width a, the light after collima- 
tion passes through a slit of width 2a and is then divided into 
equal parallel beams by refraction through a five-sided prism. 
Thus the beams are separated by a distance w which depends on 
the size of the prism. Williams has shown that with this arrange- 
ment the primary slit Si may be opened to a width Q.715w/a 
times the maximum value used in the Rayleigh refractometer, 
resulting in a considerable increase in the intensity of the fringes, 
which permits greater accuracy of measurement. 

Problems 

1. The light from a straight incandescent filament falls on two parallel 
slits separated by 0.2 mm. If the interference fringes on a screen 75 cm. 
away have a spacing of 2.2 mm., what is the wave-length of the light 
used? 

2. One of the tubes of a Rayleigh refractometer is filled with air, the 
other being evacuated. Then the second is filled with air under the 
same conditions of temperature and pressure and 98 fringes are seen to 
pass the field of view. What is the index of refraction of the air if 
sodium light is used and the tubes are each 20 cm. long? 

3. What will be the angle of tilt of the compensating plate required to 
restore the fringes to their original condition, in the preceding problem, 
if the plate has a thickness of 5.1 mm.? (Use n = 1.5 and derive an 
equation similar to that used in Experiment 10.) 

4. The interference pattern shown in Fig. 10-9 is twice the size of the 
original photograph. If the biprism was 35 cm. from the slit, and the 
photographic plate 448 cm. from the biprism, what was the wave-length 
of the light used? (NOTE: the diifraction patterns also shown in the 
photograph, on either side, are those due to the common base of the 

1 WILLIAMS, W. E., Proceedings of the Physical Society of London, 44, 451, 
1932. 



136 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. X 

prisms. By measurements of their separation, the distance d in eq. 10-9 
may be obtained.) 

5. A Fresnel biprisrn, in which the refracting angles are 2 deg. and 
the index of refraction is 1.5, receives from a narrow slit the light of the 
mercury green line, 5461 angstroms. A soap film is placed in the path 
of the light which has passed through one of the prisms, and the inter- 
ference fringes shift 3.5 fringes. What is the- thickness of the film in 
millimeters? (Assume n = 1.33 for the soap solution.) 



CHAPTER XI 
INTERFERENCE OF LIGHT DIVISION OF AMPLITUDE 

In Chap. X it was pointed out that in general there are two 
ways of producing interference of light: (A) By a division of the 
wave front into two or more sections restricted in width, which 
are later recombined, and (B) by a division of amplitude of a 
more or less extended portion of the wave front into beams which 
are afterward recombined to produce interference. The first 
of these methods has been illustrated in Chap. X by Young's 
apparatus, the Fresnel biprism and mirrors, and the Rayleigh 
refractomcter. 

1. *6olors in Thin Films. Perhaps the simplest example of 
the division of amplitude is the colors in thin films. A simple 
device showing this type of interference is a pair of plane glass 
plates pressed close together at one edge and separated by a very 
thin sheet of foil at the opposite edge, so that the enclosed 
air film is in the shape of a wedge. In Fig. 11-1 is a sketch of the 
arrangement, and in Fig. 11-16 is a photograph of fringes obtained 
with it. In the sketch the angle of the wedge is much exagger- 
ated. Also, for simplicity the changes of wave-length and direc- 
tion which take place in the glass because of refraction are ignored. 
In Fig. 11-16 the slight curvature of the fringes is due to unequal 
pressure on the ends of the plates, which were bent a little by 
cjamping. 

2. Newton's Rings. The first accurate measurements of 
interference fringes were made by Newton, although he did not 
recognize in the phenomenon of Newton's rings the superposition 
of two wave fronts. Instead, he proposed an explanation based 
on a corpuscular theory of radiation, making certain assumptions 
as to the manner in which the reflection and refraction of the 
light took place. 

The rings are obtained when two plates of glass having slightly 
different curvatures are pressed together so that they touch at one 
point. The thin wedge-shaped film of air enclosed between the 
plates provides a path difference between the two reflected 

' 137 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XI 

* 

beams (see Fig. 11-3) of at most a few waVe-lengths, so that the 
fringes may be observed with white light. If one of the glass 
plates is accurately plane and the other slightly convex and 
spherical, the fringes are concentric rings of color. 

It is perhaps less complicated to consider first the general 
case of interference in the case of films instead of proceeding at 
once to the particular case of Newton's rmcrs. Althoiiffh in 





(a) (6) 

FIG. 11-1. Interference fringes with a wedge-shaped film of air enclosed 
between two plane plates. The fine lines in the photograph are due to scratches 
on the glass plates, which were old interferometer plates. 

the following derivations the phase difference, and hence the 
interference, between beams will be referred to, it must be kept in 
mind that the interference is actually between pairs of wave 
fronts which owe their presence to a division of amplitude of the 
primary wave front incident upon the apparatus. 

Consider light to be incident in air upon a thin film of trans- 
parent medium having plane-parallel surfaces, an index of refrac- 
tion /i, and a thickness e, as illustrated in Fig. 11-2. While the 
beam reflected at A is proceeding toward G, that refracted at A 
and reflected at B must traverse the path ABA'. Hence the 
difference of path at any front as indicated by the line A'F, drawn 



SEC. 11-2] 



DIVISION OF AMPLITUDE 



139 



normal to AG and A'H, will^be (AB + BA'} - AF. But this 
is equal to 2ep cos i* where i is the angle at which the ray strikes 
the surface BB'. If, then, the two reflected rays are brought to 
a focus on a screen by means of a lens, it 
appears that there would be a maximum of 
intensity when 2eju cost' = nX, and a mini- 
mum when 2ejj, cos i 1 = (n -j- M)^> where 
n is an integer representing the number of 
waves in the difference of path. Actually, 
however, the conditions of reflections are 
not the same at A and B. At A the reflec- 

ri takes place in air from the bounding 
face of a denser medium, while at B the 
rejection is in the denser medium from a 
bounding surface of air. It has been found 
by experiment that in the former case there 
is a change of phase of IT, corresponding to 
a path difference of X/2, while in the latter 
case there is no change of phase, so that the 
situation is exactly opposite to that stated above, and the beams 
AG and 4/# will reinforce each other to produce a maximum when 

(11-la) 



(11-16) 




K 



cos i' (n -f- 
while they will produce a minimum when 

2eiJ. cos i' = nX. 



On the other hand, since the transmitted rays BJ and B'K 
suffer no difference of phase, they will reinforce each other when 

2e/i cos i' = wX, 



and counteract each other when 



cos 



(n -f 



(11-26) 



In the case of Newton's rings, eqs. 11-la and 11-16 will hold 
for reflected light, provided /* is put equal to unity, since the 
two interfering reflections take place with opposite phase exactly 
as in the case described above. 

The radius of any ring may be found as follows: Let r (Fig. 
11-3), be the radius of curvature of a curved glass surface AOM 
which rests upon a plane glass surface OB. Then, since LA is 



140 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XI 



nearly the same as OB, and AB is nearly the same as e, the 
separation between the two surfaces at B, it follows that 



_ ( r _ 6 )2 



-f 



where p is the radius of the fringe caused by the interference 

of the two beams, one reflected 
from the upper surface at A, the 
other from the lower surface at B. 
Since e must be small compared to 
the other dimensions for the third 
condition for interference to bo 
fulfilled, 

p 2 = 2re 

to a sufficient degree of apprwria- 
tion. But by eq. 11-1, when /u is 
unity 

(n 




2e = 



COS 7, 



FIG. 11 -3. -Illustrating the for- f or a bright .fringe, and hehc< 

mation of Newton s rings. The . 

curvature of AOM is exaggerated; 

actually i at A and B would be p 2 = -- '.j|-*' ..y 

almost the same. 



COS 



For small angles i\ cos i' is approximately unity, aj 
of the fringe is given by 



where n has the values 0, 1, 2, 3, etc., for the first , 
etc., rings, respectively. The radius of a dark fringi 
given by 




P 



(11-5) 



Since for n = 0, by eq. 11-5, p = 0, for reflected light there will 
be a dark spot at the center of the fringe system. Also, the 
radii of the dark fringes are proportional to the square roots of 
whole numbers. Similarly, for the transmitted light, there will 
be a bright spot at the center, and the radii of the bright rings 
are proportional to the square roots of whole numbers. Thus 
the interference patterns produced by reflection and trans- 
mission are complementary. 



SBC. 11-2] DIVISION OF AMPLITUDE 141 

A photograph of Newton's rings obtained with monochromatic 
light is shown in Fig. 11-4. 

If the upper plate is made of glass of a smaller index of refrac- 
tion than the lower one and a liquid of intermediate index is 
placed between them, the pattern obtained by reflection will bo 
complementary to that obtained with an air film between the 
plates, since at each interface the reflection will take place in a 
medium of a given index at the bounding surface of one of higher 




FIG. 11-4. Photograph of Newton's rings obtained with an uncemented 
achromat. The white spot occurs at the center because the two surfaces were 
slightly separated there. 

index of refraction, and in both cases a change df phase of ir 
will occur, 

The pattern obtained by transmission is not as easy to see 
as that obtained by reflection, since the light transmitted is 
much greater than that reflected, resulting in a background of 
light against which the interference pattern is dimly observed. 

It is desirable that the surfaces for producing Newton's rings 
be clean and free from oxidation. In order that the best results 
may be obtained it is necessary that the glass surfaces be freshly 
ground and polished, since the films which develop with age are 
not removable by ordinary means. 1 

1 See, for instance, a brief paper on this subject by W. W. Sleator and 
A. E. Martin in the Journal of the Optical Society of A im-rica 24, 29, 1934. 




142 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XI 

The colors sometimes seen in films of oil on wet pavements, in 
soap bubbles, in fractures along cleavage planes of crystals, etc., 
are all analogous to Newton's rings, since they are due to inter- 
ference between wave fronts reflected from the surfaces of thin 
films. For this reason they may be observed with white light, 
since the path difference is small. All such interference fringes 
belong to class B. 

'3. Double and Multiple Beams. Any apparatus for producing 
interference by a division of amplitude has as one of its principal 
features a surface, illustrated in Fig. 11-5, which reflects part 
of the light and transmits as much of the remainder as is not 

absorbed. The two parts of the amplitude thus 
/ divided must be recombined later in such a man- 
' ner that the conditions for interference stated in 

arace ^^ ^^ ^^ fulfilled. The particular mariner 
in which this recombination takes place depends 
on the type of instrument used. In some, the 
FIG. 11-5. beams are recombined without further subdivi- 
sion. The best known and most useful instrument of this type 
is the Michelson interferometer. In others, a second reflecting 
surface is placed parallel to the first as illustrated in Fig. 11-6. 
If this surface is also partly transmitting, it is evident that there 
will be two sets of parallel beams, one on either 
side of the pair of surfaces. Moreover, between 
the successive beams in either set there will be 
a constant difference of path. When the beams 
in either or both sets are collected by image 
producing mirrors or lenses, an interference 
pattern will appear at the focal plane of the 
system. It will be seen in the discussion of FlG - u " 6 ' 
the Fabry-Perot interferometer, which best exemplifies this type 
of instrument, that this superposition of many beams results 
in a great increase in the resolving power of the instrument. 
It should be remarked, however, that the term "resolving 
power" in its broadest sense does not mean merely the ability 
to produce on a screen or on the retina of the eye two actual 
and distinct images of the object. It may mean the ability to 
produce phenomena from which may be deduced the existence 
of two objects whose relative intensities and separation may be 
found. Later we shall see that in this sense of the term the 




SEC. 11-4] 



DIVISION OF AMPLITUDE 



143 



Micholson interferometer possesses theoretically unlimited resolv- 
ing power. 

All instruments which make use of the principle of division of 
amplitude may be considered as modifications of the Michelson 
or Fabry-Perot interferometers, which are accordingly described 
in /the following pages in some detail. 1 _ v,,/. v 

. The Michelson Interferometer. VThe many forms of this 
instrument are alike in that the amplitude of a wide beam of 
light is divided into two parts by means of a semitransparent 
plate. The form which Michelson adopted as most useful for 
a variety of purposes is illustrated in Fig. 11-7. Here the 
division of amplitude is effected by the plate A, a plate of glass 




Fia. 11-7. 

with parallel surfaces, one side of which is usually lightly coated 
with metal so as to divide the intensity of the beam into two 
equal parts. Half the light is thus transmitted to the plane 
mirror C, the other half reflected to the plane mirror B. The 
plane parallel plate D is cut from the same plate as A but is not 
metallically coated. It is placed between A and C, parallel 
to A, so that the optical paths ABA and AC A contain the same 
thickness of glass. This is important whenever observations 
are made of fringes due to light of many wave-lengths, as in 
white light, since the index of refraction of the glass varies with 
the wave-length. -* 

*'The interference pattern is observed at J0J Here the light 
from B and C appears to have originated in two virtual image 

1 For a description of many types of interferometers and a good biblio- 
graphy see Williams, "Applications of Interferometry," Dutton. 



144 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. Xl 



planes situated in the neighborhood of B. We may consider one 
of the virtual image planes to be Mi (Fig. 11-8). Let Mi be 
the plane which replaces mirror B. If Mz is, likewise, the plane 

. which replaces mirror C in the field of view, 
-M 2 then the virtual image due to the light from 
~ M * C must be in a plane Mi which makes an 
FIG. 11-8. angle with Mi twice that between Mi and 



1 Thus for purposes of analysis the mirrors B and C are 
replaced by two virtual image planes M i and Mi, and the interfer- 
ometer is considered as a pair of plane wave fronts with an air space 
between them. If the distances from A to B and A to C are 
not equal, and if B and C are not at right angles, these wave 




FIG. 11-9. 

fronts will be as shown in Fig. ll-9jmt is desired to find the 
character of the fringes formed at some point P. The first 
step is to find the path difference between the two virtual wave 
fronts. 2 

The following notation will b* used: 

D = the distance from a point P, where the interference 
fringes are formed, perpendicular to the planes Mi and 
Mi, which must for the production of fringes be at such a 
small angle that they can be considered to have a com- 
mon perpendicular. 

2<f> = the small angle between the surfaces, in a plane perpen- 
dicular to their line of intersection. The plane in which 

1 Since, when a reflector is turned through an angle, the reflected beam is 
turned through twice that angle. 

2 The derivation given here is essentially that presented by Michelson in 
Philosophical Magazine, (5), 13, 236, 1882. 



SEC. 11-4] DIVISION OF AMPLITUDE 14ft 



this angle lies will depend on the adjustment of 
mirrors A and B. 
d = the angle between the perpendicular D and the line 

joining P and a\. 

i = the projection of the angle 5 on a plane containing 2<p. 
= the projection of the angle 5 on a plane perpendicular 

to that containing 2<p. 

A = the difference of path between the distance a\P and 6iP. 
2t = the distance a b . 
2t = the distance a\b\. 

The planes Pb and ai&i are two parallel lines which define a 
plane in which the angle 8 lies; hence, 

1 A = aibi cos 5 = 2t cos d. (11-6) 

~ 

But 2t = 2to + a\c tan 2<p, or, since a\c = D tan i, < 

t to + D tan <p tan i 

to a sufficient degree of approximation since the angles are small. 
Substituting this value of t in eq. 11-6, 

A = (2*o + 2D tan <p tan i} cos 6. (11-7) 

But 

D 



tan 2 i -f 



cos 5 = 



$ 
Hence, 



A = *- (11-8) 

VI + tan 2 i + tan 2 

'^> 

Thus we see that the path, and hence the phase, difference 
between the two beams a\P and biP may vary over the area of 
the wave front contributing to the fringes at P, and the phenome- 
non of interference may be obliterated. If the sizes of the 
angles <p, i, and 8 are restricted sufficiently so that the maximum 
value of A is X/2 or less a single phase will predominate and the 
fringes will be distinct. In most cases the pupil of the eye places 
a sufficient restriction on i and 0, provided it is at a suitable 
distance from the interferometer, so that the fringes are easily 
seen. Sometimes the use of a pin jiole in front of the eye will 
improve the visibility of the fringes 



146 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XI 

5. The Distinctness of the Fringes. The fringes will be most 
distinct when dA/d0 and d&/di are both zero. Imposing on 
eq. (11-8) these two conditions gives 

D - fc*ELi, (ll_9) 

tan <p ^ ' 

for the distance between P and the position of the surfaces for 
which the fringes will be most distinct. An examination of 
eq. 11-9 shows that if U = then D = 0, and the fringes will be 
best at the surface of B (Fig. 11-8). This means that if the eye is 
placed at normal reading distance from the mirror J5, the fringes 
will appear distinct when the lengths of the optical paths in the 
two arms of the interferometer are the same. Equation 11-9 
also says that when i is zero, D is zero, which is another way of 
saying that the area of the wave front is sufficiently restricted in 
a direction perpendicular to the intersection of the two wave 
fronts so that no troublesome confusion of phases exists. If 
<p is zero, the wave fronts are parallel, i.e., the mirrors are at 
right angles; if also to is zero, the optical paths in the two arms of 
the interferometer are the same and also the mirrors are per- 
pendicular, and over the entire field the two wave fronts will 
cancel each other. If <p and i have the same sign, D is positive, 
and the fringes are formed in front of the mirror B', if they 
ha^ve opposite sign, D is negative and the fringes lie behind B. 

6. The Form of the Fringes. Any point on the plane where 
thlf fringes are formed may be described by the equations 

x D tan i 
y = D tan ( 

Substituting these values in eq. 11-8, we obtain the general 
equation for the form of the interference fringes, 

Ay = (4D 2 tan 2 <p - A 2 )z 2 + (StoD 2 tan <p)x + D 2 (W - A 2 ). 

(11-11) 

An analysis of this equation shows 1 that the fringes take the 
forms of straight lines, circles, parabolas, ellipses, or hyperbolas, 
depending on the values assigned to A and <p. The complete 
theory will not be discussed here; certain details are, however, 
worthy of attention since they bear directly upon the successful 

1 SHEDD, JOHN C., Physical Review, 11, 304, 1900; also Mann, "Manual of 
Optics," Ginn. 



SEC. 11-71 DIVISION OF AMPLITUDE 147 

use of the instrument. For A = 0, eq. (11-11) becomes 

* = T-^-> (11-12) 

tan <f> v ' 

which is the equation of a straight line. This is the central 
fringe of the system of fringes obtained with a white-light source. 
Those on either side of it, corresponding to A very small, will be 
curved in opposite directions on either side of the central fringe, 
although the curvature is not noticed for the few fringes which 
occur with a white-light source. The curvature is scarcely 
noticeable with a monochromatic source within the relatively 
small area of the field of the instrument unless A is large enough 
to correspond to about 100 fringes from the central fringe. By 
moving the mirror B back and forth rapidly about the position 
for A = 0, the change of curvature may be detected. This 
maneuver constitutes one method of finding approximately the 
center of the fringe system. With the limits of the position for 
B thus determined, white light may be substituted for the mono- 
chromatic source, and the mirror moved very slowly until the 
white-light fringes come into the field of view. 
If (f> = 0, eq. 10-11 becomes 



2 - A 2 ) 

x z + y- = - --- , (11-13) 



which is the equation of a circle. Hence, when the virtual source 
images are parallel, the fringes will be circular in form5j As A 
becomes small, the diameters of the circles will become large, 
until for A = 0, the entire field will be either dark or bright. In 
theory it will be dark for A = 0, since the reflection of the 
two beams at the dividing plate A introduces a difference of 
phase of ir, and a difference of path of X/2. Actually, however, 
a field entirely dark is difficult to observe since very slight irregu- 
larities in the metallic coats, lack of planeness of the glass 
surfaces, or inhomogeneity in the glass may have an effect 
which is of maximum observability for this adjustment of the 
interferometer. The field will in general be dark, with irregular 
streaks and patches of light showing. 

7. The Visibility of the Fringes. Visibility Curves. Although 
the Michelson interferometer was originally designed as a refrac- 
tometer to measure the relative difference of path introduced 
into the two arms of the instrument by a change in the medium. 



148 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XI 

it has been used with great success, especially by its inventor, 
in the analysis of_cpmplex spectral radiations. The method used 
depends Tipon the fact, "already mentioned, that in its broadest 
sense the term "resolving power" does not necessarily imply 
the actual separation of the images of two sources, but rather the 
production of a pattern of light in the images which may be 
interpreted as indicating the presence of two separate sources 
with a determinable separation and ratio of intensities. 

In the Michelson interferometer the entire pattern of light 
here referred to is in most cases not observable in the field of view 
at the same time but must be examined while the path difference 
between the two arms is changed. In order to outlinc^the 
method, we may first consider the difference in the appearance 
of the fringes obtained with white light and with monochromatic 
light. Suppose mirrors A and B (Fig. 11-7) are placed so 
that the path difference at the middle of the field is zero. We 
may then indicate the two virtual wave fronts producing the 
interference by two crossed lines as shown in Fig. 11-10. If the 
source is white light, there will be a black fringe corresponding 
to the intersection, since at the half plate A (Fig. 11-7) one 
reflection is from a bounding surface of glass, the other from 
one of air, so that a difference of path of X/2 is introduced between 
the two interfering beams. On either side of this will be fringes, 

alternately light and dark. These will be 
colored, since at any position, such as X (Fig. 
11-10), in the field of view where the difference 
of path is such as to produce a dark fringe 
for one wave-length, there will be light for 
FIG. 11-10. other wave . lengt j ls xhe result will be that 

due to the superposition of an infinite number of fringe 
systems, one for each wave-length emitted by the white light 
source, all of which have different spacings. In consequence, 
only about a half dozen fringes will be seen on either side 
of the middle dark fringe, and beyond this range the field will be 
uniform. Another way of making the last statement is to say 
that the visibility 1 of the fringes will diminish gradually to zero, 

1 The visibility of the fringes is defined as the ratio ,"" , /" > where 

'max T" 'mln 




mln 

is the maximum intensity of the fringe system, and /,, is the minimum 
intensity. 



SBC. 11-7] 



DIVISION Ob' AMPLITUDE 



149 



so that beyond a half dozen on either side of the central minimum 
none will be seen. On the other hand, if .strictly ..mongchrgmatic 
light is used, there will be no dimi- 
nution of the visibility of the fringes, 
no matter how far away from the 
central dark fringe they are exam- 
ined. These details are illustrated 
with a fair degree of exactness in 
Fig. 11-11, in which (a), (6), and (c) 
are photographs obtained with red, 
green, and blue monochromatic 
radiations, respectively, and (d) is 
the photograph of the white-light 
fringes due to the superposition of 
all the radiations from a white-light 
source. Although the range of sen- 
sitivity of the eye is not the same 
as that of the photographic plate, 
Fig. 11-1 Id approximates closely 
the visual appearance of white-light 

frin g es - Center 

Figure 11-12 illustrates the Fro. ll-ll Interference fringes 
effect produced by narrowing the with a Mi <> h elson interferometer. 

spectral range of white light. In a are shown the fringes due 
to the entire range of wave-length in a white-light source, while 
in 6 are shown those obtained when a filter is interposed, which 





(a) (ft) (c) 

FIG. 11-12. (a) White light fringes. The arrows point to the central dark 
fringe. (6) Fringes with white light through a filter which transmitted a band 
of about 100 angstroms. The arrows point to the central dark fringe, (c) 
Fringes with the green line of mercury, 5461 angstroms. 

permits the passage of a band of light of only about 100 ang- 
stroms. In Fig. ll-12c are shown the fringes, with the same 



150 LIGHT: PRINCIPLED AND EXPERIMENTS [CHAP. XI 

adjustment of the interferometer as in a and 6, from the green 

mercury line, whose wave-length is 5461 angstroms. 

If the light incident upon the interferometer is composed of 

two radiations, the visibility of the fringes will pass through 
alternations whose spacing will depend upon the ratio of the two 
wave-lengths. If this ratio is large, the alternations will occur 
rapidly, as illustrated in Fig. ll-13a and 6. In a is shown the 
effect due to the superposition of the fringes of the two mercury 
lines 5461 and 4358 angstroms, and in b is shown the effect due 
to the mercurv lines 5461. 5770. and 5790 ano-stroms. The 




(a) (b) 

FIG. 11-13. Interference fringes (a) with X4358 and X6461; (6) with X5461 and 
the two yellow mercury lines X5770 and X5790. 

last two are so nearly alike that in the field of view they have the 
effect of a single radiation. 

Thus far in the discussion of visibility, the effect due to indi- 
vidual spectral lines has been treated as though each such radiation 
were monochromatic. Actually, however, there is no such thing 
as a completely monochromatic radiation, although in some the 
range of wave-length dX is extremely small. Owing to circum- 
stances which depend on the nature of the radiating atoms or 
molecules and the conditions in the source, even the most nearly 
monochromatic radiations have a width rfX, so that with sufficient 
difference of path introduced between the two arms of the inter- 
ferometer the visibility of the fringes will drop to zero. More- 
over, most so-called single spectral lines, such as the mercury 
lines mentioned in the last paragraph, are composed of several 



SEC. 11-7] DIVISION OF AMPLITUDE 151 

individual lines whose difference of wave-length is so small that 
none but the highest resolving power will make it possible for 
them to be observed directly as separate lines. The Michelson 
interferometer possesses theoretically unlimited resolving power; 
although it does not enable the observer to see this fine structure 
of spectral lines directly, the interpretation of the alternations of 
visibility just illustrated makes it possible to determine the 
presence and character of the fine structure. 

No satisfactory method has been developed for determining 
accurately changes in visibility by any but visual means. Conse- 
quently, Michelson's method of analysis of spectral lines by 
visibility curves, while it was the first to yield the structure of 
many important radiations, has not progressed beyond the initial 
stages developed by its inventor. Other instruments of high 
resolving power have taken its place, although the results 
obtained by Michelson have in many cases not been surpassed in 
accuracy. His method consists essentially in plotting the 
visibility graphically as a function of the difference of optical 
path between the beams traversing the two arms of the inter- 
ferometer. This graph, however, may be regarded as the result- 
ant intensity graph of a number of separate intensities. By the 
use of specially designed mechanical analyzers, these components 
are found, and the wave-length ratios and relative intensities of 
the corresponding individual lines determined. A few of Michel- 
son's visibility curves, taken from his published papers, are 
shown in Fig. 11-14. 

By means of this method, Michelson was able to show that the 
red radiation from cadmium vapor was most nearly mono- 
chromatic of all those that he examined. Accordingly he 
used it as a primary standard of wave-length for comparison 
with the standard meter, using interference methods. 1 This 
comparison was carried out first by Michelson, and later by 
Benoit, Fabry, and Perot with the Fabry-Perot interferometer, 
confirming Michelson's measurement. 2 The value of the wave- 

1 " D6termination experimeiitale de la valeur du metre en longueurs d'onde 
lumineuses." Translated from the English by Benoit. The details are 
described briefly in Michelson, "Light Waves and Their Uses," and Michel- 
son, "Studies in Optics," both published by the University of Chicago Press. 

2 BENOIT, FABRY, and PEROT, Travaux et memoirs Bureau international, 
11, 1913. 



152 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XI 



length of the red cadmium line thus obtained in terms of the 
standard meter in dry air at 15C. and 760 mm. Hg pressure is 
6438.4696 angstroms. This value has been accepted by inter- 
national agreement as a primary standard of wave-length. 



(a) 



(b) 



(c) 




0.1 20 40 60 80 100 120 140 160 180200220240mm. 
A ff 




0.1 0.2 20 40 60 80 100 120 140 160 180 200 220 240 260 280mm. 
A B 




O.I 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320mm. 
A B 

1.0 



0.5 




0.1 



0.2 
A 



0.3 0.4 



10 



20 



30mm 



Fio. 11-14. Michelson's visibility curves. In each ease graph B shows the 
variation of visibility of fringes with path difference in millimeters, and A shows 
the interpretation of B in terms of intensity distribution in the spectral line used 
(a) The red cadmium line (primary standard) 6438.4696 angstroms; (b) the 
sodium lines 5890 and 5896 angstroms; (c) the mercury line 5790 angstroms; and 
(rf) the red hydrogen line, H, 6563 angstroms. 



By the use of visiblity curves, Michelson was also the first to 
show that the red hydrogen line, 6563 angstroms, is really a very 
close double. His result is in good agreement with those obtained 
later by the use of instruments of direct resolving power, and 
with the structure of the line deduced theoretically by the applica- 
tion of the Quantum theorv to the analysis of spectra. 



SEC. 11-8] 



DIVISION OF AMPLITUDE 



153 



8. Multiple Beams. The Fabry-Perot Interferometer. In 

general, the superposition of multiple beams results in higher 
resolving power than is obtained with a double-beam instrument. 
This is seemingly in contradiction to the fact already stated, 
that the Michelson interferometer has theoretically unlimited 
resolving power. The higher resolving power obtained with the 
Fabry-Perot interferometer, however, is due to a sharpening of 
the maximum of intensity in the interference pattern to the 
point where the existence of two separate images may be observed 
directly, while with double beams only, an analysis of the visi- 
bility of the interference fringes is required. 1 There is this 
difference also: While the resolving power of the Fabry-Perot 




FIG. 11-15. 

interferometer is limited by the reflecting power of the surfaces, 
the limitation in the case of the Michelson interferometer is not 
in the instrument itself, but in the ability of the observer to 
distinguish and interpret correctly the variations in the visibility 
of the fringes. 

PThe Fabry-Perot interferometer, illustrated in Fig. 11-15, is 
constructed of two plates, usually of glass or quartz, having their 
faces accurately plane, and mounted so that the adjacent surfaces 
are parallel. These parallel surfaces are coated with a metallic 
film capable of transmitting part of the light and reflecting a high 
proportion of the remainder. Consider light of a single wave- 
length, X, incident upon the metallic coating of plate A, at an 
angle <p. Part of it is reflected and part is transmitted to surface 
B. At this latter surface, part of the incident light is reflected 
and part transmitted. Of the part reflected back and forth 
between the two surfaces, a fraction is transmitted through B at 

1 An analogous comparison may be made of the diffraction grating and 
the Michelson stellar interferometer. For details, see Chap. XII. 



154 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XI 




each incidence upon it. There are similar sets of reflections 
due to the surfaces of the plates on which no metal is deposited, 
but these surfaces have relatively low reflecting power. It is 
customary to make each of the plates slightly wedge-shaped, so 
that its two surfaces are at a small angle. This causes any 
reflections at the outer, uncoated surfaces to be thrown to one 
side so as not to be superposed upon the pattern of fringes 
obtained with the inner metal surfaces. This wedge shape 
introduces a slight amount of prismatic dispersion, but not 
enough to cause serious difficulty* For angles of incidence 

greater than zero, each beam under- 
goes a small sidewise displacement 
owing to refraction, but this is the 
same in both plates for all beams 
having the same angle of incidence, 
and so may be neglected. 

We may thus consider the inter- 
ferometer to be essentially a pair of 
parallel surfaces of as high reflect- 
ing power as possible. By reference 
to this simplified concept which is illustrated in Fig. 11-16, we 
may readily see that 

AD = p\ = 2d cos <p, (11-14) 

in which p is the number of wave-lengths in the common differ- 
ence of path of consecutive rays such as Ei, Ez, etc., and d is the 
separation of the surfaces. We may then call p the order of 
interference between the successive beams E\, E z , etc. It should 
be emphasized that the focal poinkof these parallel beams will be 
the principal focus of the eye lens, in case of visual observation 
of the fringes, or of the projecting lens, as shown in Fig. 11-15, 
regardless of the manner in which the original beam of light is 
projected upon the interferometer. The plane of incidence 
represented by the page in Fig. 11-15 is one of an infinite number, 
all containing the normal to the reflecting surfaces, hence there 
will be a circle of focal points for each angle of incidence <f>. If p 
is a whole number, the difference of path between successive 
elements will be an integral number of wave-lengths, and the 
amplitudes of the successive beams will add to give a maximum 
of intensity in the form of a circular fringe. Since there will be 



SEC. 11-8J DIVISION OF AMPLITUDE 155 

for any wave-length several values of <p for which p will be a whole 
number, there will correspond to each wave-length a number of 
concentric circles of maximum intensity! 

At the center of the pattern, the intensity will depend on 
the difference of path for <f> = 0. For this case, eq. 11-14 becomes 

PX - 2d, (11-15) 

where P is used to indicate the order of interference at the center 
of the ring system, while p is used to indicate the order of inter- 
ference for a bnghtfrinee^ExceDtinanoccasional instance, 




Fio. 11-17. Fabry-Perot fringes of the mercury line 6461 angstroms. 

P is not a whole number, while p is always a whole number. 
Provided all wave-lengths undergo the same change of phase on 
reflection from the metallically coated surfaces, we may assume 
d to be constant, whereupon 

PiXi = P 2 X 2 = = constant, (11-16) 

so that if the ratios of the P's can be found, the ratios of the wave- 
lengths may be calculated. If one of the observed radiations is 
either the primary standard, 6438.4696 angstroms, or else a 
suitable secondary standard of wave-length, the other wave- 
lengths may all be found with a high degree of accuracy. The 
use of the Fabry-Perot interferometer for the comparison of 
wave-lengths with primary or secondary standards has been 
adopted by international agreement for the establishment of 
wave-lengths of spectral lines throughout the visible, ultraviolet, 



156 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XI 

and infrared spectral regions. For secondary standards, the 
spectra of iron, copper, and neon are principally used. 

The use of the Fabry-Perot interferometer in the measurement 
of the length of the standard meter in terms of the wave-length 
of the red cadmium line has already been referred to in Sec. 11-7. 
The instrument may also be adapted to a number of other uses, 
probably the principal one at the present time being the examina- 
tion of the fine structure of spectral lines. l A photograph of the 
system of fringes of the green mercury radiation, 5461 angstroms, 
is shown in Fig. 11-17. The composite structure of the line, 
which can be found also by the visibility-curve method with the 
Micjielsori interferometer, is shown very well. 

. Intensity Distribution in Fabry-Perot Fringes. From a 
consideration of Fig. 11-16, it is evident that the rate at which the 
intensity of the successive parallel beams E\, Ez, E 3) etc., 
decreases depends upon the reflecting and transmitting powers 
of the metallically coated surfaces. Let Q and R represent 
respectively the fractional parts of the incident light intensity 
transmitted and reflected at each of the surfaces. Then the 
transmitted beam E\ will have an intensity Q 2 and an amplitude 
Q; beam E z will have intensity Q 2 R 2 and amplitude QR; beam # 3 
will have intensity Q^R* and amplitude QR 2 , and so on. The 
amplitude of the nth beam will be QR n ~ l . Neglecting the small 
change of phase on reflection which takes place at the surface 
of the metal, the constant difference in phase between successive 
beams is 

= 8 ' 

A 

The disturbance at any point on the incident wave front may be 
represented by 

S a cos 2-^ = a cos wt. 

It is possible, however, to use exponential instead of trigono- 
metric expressions with some shortening of the labor involved. 
Since e** = cos at + i sin at, the disturbance may be repre- 

1 For a fairly complete discussion of the uses of the Fabry-Perot inter- 
ferometer, and an excellent bibliography, see Williams, "Applications of 
Tnterferometry." 



SEC. 11-9] DIVISION OF AMPLITUDE 157 

sented by the real part of ***. If the difference of phase between 
the successive transmitted beams E\, / 2 , etc., is 6, the total 
amplitude of the sum of the beams at any instant will be the 
real part of 

2 = Qe 1 ^ + QRe**-'* + Q/2V ( '- 2 *> + (11-18) 
This can be put in the form 

(1M9) 



But the amplitude factor 

Q C 



1 - Re~* 1 - R cos 6 + iR sin 5 

_ ~ 1 R cos 6 iR sin 6 
_ ^ _________, 

an expression in which the numerator is of the form X iY, 
where X = Q(l R cos 5) and Y = QR sin 6. Since this 
represents the amplitude of the superposed beams, the intensity 
is given by 

; (1 - R cos 6) 2 + R z sin 2 6 
(1 --~272"c()s"6"4-~_2 2 ) 2 

Q 2 



(1 - 2R cos 6 + /2 8 ) 



1 + 2/2(1 - cos 6) - 2# + R- 

Q2 

(1 - RY + 4/2 sin 2 (6/2) 
Q 2 1 



(1 - RY 4# /Y 

> ' -f I AjLt 1 Of 1 ' 

1 4. _ r>N - . sm- ( ^ 



(11-20) 



When 6 = 0, 2ir, 4?r, etc., sin 2 (6/2) = 0, and the maximum 
intensity of the fringes is Q 2 /(l /2) 2 ; when 6 = T, Sir, 5ir, etc., 
sin 2 (6/2) = 1, and the minimum intensity of the fringe system 
is Q 2 /(l + /2) 2 . It will be seen that the intensity never drops 
to zero although it may become very small. 
The visibility of the fringes is defined as 

7 T 

F-* mftx * mjp 

- f -TTT" 

* max i^ * min 



158 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XI 

Hence for the Fabry-Perot fringes, the visibility is 

V = -y (11-21) 



and thus depends only on the reflecting power of the metal 
surfaces, and is independent of their transparency! 

10. Resolving Power of the Fabry-Perot InteTferometer. By 
uuicrentiation, from eq. 11-15, we obtain 

P d\ + X dP = 0, 

from which it follows that the resolving power is 

X P 



or, the resolving power, defined as the ratio of the wave-length 
to the smallest difference of wave-length which may be detected, 
is equal to the order of interference at the center of the ring 
system divided by the smallest change of order dP which can be 
detected. Actually, since the value for p is different from 
that for P, for any given wave-length, by only a small number, 
provided one considers a bright fringe only a few rings outside 
the center, the actual measurement of dP may be more easily 
made on a fringe near the center instead of at the center itself, 
since at the center of the pattern the width of the rings is so large 
as to render estimates of intensity variation in them difficult. 
This point is taken up in detail in Sec. 11-11. 

The value of dP may be found from eq. 11-20. Consider two 
adjacent bright fringes in the interference pattern, belonging to 
two wave-lengths, X and X + d\ between which the difference 
of order dP, corresponding to their difference of wave-length d\, 
is to be found. It will be shown in the chapter on Diffraction 
that, according to. an arbitrary criterion established by Rayleigh, 
two images are said to be just resolved when the maximum of 
intensity of one of them corresponds in position with the first 
minimum of intensity of the other, as illustrated in Fig. 11-18. 
This criterion, which agrees very well with experimentally 
determined measures of limit of resolution in optical imagery, 
was originally set up with regard to spectral-line images produced 
with diffraction gratings, and may be considered to hold suffi- 
ciently well in the present case. The intensity for either of the 



SEC. 11-10] 



DIVISION OF AMPLITUDE 



159 



adjacent bright fringes in the Fabry-Perot ring system may be 
called /,, and by the last section is equal to Q 2 /(l jR) 2 , while 
the intensity at the center of the pattern shown in Fig. 11-18 is, 
for either image, given by eq. 11-20. Consequently we may write 



1 



1 




(11-23) 



It can be shown from diffraction theory that in case the images 
are equal in width and intensity and symmetrical, the intensity 
at a point c 2 , midway between the images, will be 8/Tr 2 , or about 




- dP H 

FIG. 11-18. 



0.81, times the maximum of either, so that the intensity of each 
image at the point c 2 is 0.405 times the intensity of either at its 
maximum. Thus we may write 

J = 0.405. 



Substituting this value in eq. 11-23, there results 

2 M _ - 0.405) (1 - 
81 \2/ 



It should be kept in mind, however, that the minimum inten- 
sity of the fringes, given by Q 2 /(l + #) 2 , never drops to zero, 
although for heavy metallic coats on the interferometer surfaces 
it may become so small that it is negligible for visual observations 
and correct photographic exposures. Also, it is not always true 



160 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XI 

that the fringes of two radiations to be resolved have even approx- 
imately the same intensity; consequently the Rayleigh criterion 
does not hold with great rigor. Moreover, the Fabry-Perot 
fringes are not symmetrical, but are unsymmetrically widened 
toward the center of the ring system. Since taking these excep- 
tions into account would require a greater departure from simple 
theory than is justified in obtaining an expression for resolving 
power, which, after all, can only agree approximately with any 
observed value, we may disregard them. Then in the present 
case dP in eq. 11-22 may be said to correspond to a change in 
phase of IT, since the difference of order of unity between two 
fringes corresponds to a difference of phase of 27r. Hence at the 
Rayleigh limit of resolution, 5 = TT dP. Substituting this value 
of 5 in eq. 1 1-23, we obtain 



6 = 2 sin- 1 



0.595(1 - 




1.627? 
so that the resolving power is 

X P PTT 



-H = dP, 



d\ (IP 



. , 0.367(1 - 
2wn ~ft 



1 



From this equation the theoretical resolving power of the Fabry- 
Perot interferometer may be calculated. The negative sign in 
front of the right-hand member may be disregarded, as it means 
simply that a positive increase of wave-length corresponds to a 
negative change of order dP. For a wave-length of 5,000 ang- 
stroms and a mirror separation d of 10 mm., it follows from eq. 
1 1-15 that P is 40,000. From eq. 1 1-24 are calculated the resolv- 
ing powers shown in the following table: 

Reflecting Power, Per Cent Resolving Power 
50 139,600 

75 349,200 

90 1,047,200 

This shows that the resolving power increases very rapidly 
with the reflecting power of the metallic coating. Only the best 
metallic coats have reflecting powers of 85 per cent or better, 
and not all metals are satisfactory for the purpose. Those 
most useful for both the visible and near ultraviolet are alumi- 



SEC. 11-11] DIVISION OF AMPLITUDE 161 

num, chromium, platinum, gold, nickel, and silicon. For the 
visible only, silver is very useful, but it possesses a band of almost 
complete transmission in the region of 3300 angstroms. Until 
recently it has been difficult to obtain uniform deposits, but the 
development of the modern evaporating process, outlined in 
Appendix V, has resulted in the production of deposits which are 
not only more uniform but more durable. In addition, the 
evaporating process has made it possible to obtain highly reflect- 
ing coats of metals not obtainable by earlier methods. The 
most useful metal for all-round purposes is probably aluminum, 
which is a good reflector over practically the entire available 
range of optical spectra and retains its reflecting power for very 
long periods. 

11. The Shape of the Fabry-Perot Fringes. In the last 
section it was stated that the fringes obtained with this inter- 
ferometer are not symmetrical about their maxima. This may 
be shown in the following manner: Dividing eq. 11-15 by eq. 
11-14, 

- = = i- /7>v (11-25) 

p cos (p cos (a/2) 

in which a is the angular diameter of the pth fringe. The cosine 
term may be expanded into a series: 



COS 1-^1 = 1 



2 ~ 4 -2! 16 -4! 



For observations made with a sufficiently small, only the first 
two terms of the series are significant, hence eq. 11-25 may be 
written 

P = 2-j (11-26) 



If D is the linear diameter of a fringe, and F is the principal 
focal length of a lens or mirror used to focus the fringes, then 
a = D/F. Hence eq. 1 1-26 may be written 




For a given fringe p is a constant, so that differentiating eq. 



162 LI$HT: PRINCIPLES AND EXPERIMENTS [CHAP. XI 

11-27 with respect to D, we obtain 

dD 



On substitution of this value for dP in eq. 11-22, it follows that 



d\ dP p D dD 

whence 

dD 



d\ ~p\D ' 

For any fixed separation of the interferometer surfaces the ratio 
P/p is constant for a given X and is practically equal to unity 
provided a fringe not too far from the center is taken. We may 
therefore write 

- 5 <* 

where K is a constant depending on the wave-length and the 
principal focal length of the projecting lens or mirror. Equation 
11-28 says that the change in diameter of a fringe with wave- 
length is inversely proportional to the diameter of the fringe. 
For fringes with very small diameters, i.e., for fringes which lie 
very close to the center of the system, the change in D with small 
changes of X will be very large. This means that the bright 
fringes in the pattern, for a single wave-length, will not be 
symmetrical in shape, but will be unsymmetrically broadened 
toward the center and sharper on the outer edge. Hence in 
determining the wave-lengths of spectral lines, it is desirable to 
avoid the use of the rings close to the center of the pattern 
unless great care is taken to set accurately on the maximum of 
intensity of the fringes rather than on the geometric center. 

Problems 

1. Describe a method by which Newton's rings could be used to 
determine the ratio of two wave-lengths 1000 angstroms apart, say 5000 
and 6000 angstroms. 

2. Between the convergent crown and divergent flint glass elements 
of an uncemented achromatic doublet Newton's rings are formed. When 
seen by reflection through the flint element there is a dark fringe at the 
center, and the fourth bright fringe has a radius of 1.16 cm. If the 



DIVISION OF AMPLITUDE 163 

radius of curvature of the crown glass interface is 50 cm., and the inci- 
dent light is nearly normal, what is the radius of curvature of the flint 
glass face next to it? Assume a wave-length of 5500 angstroms. 

3. A Michelson interferometer is adjusted so that white light fringes 
are in the field of view. Sodium light is substituted and one mirror 
moved until the fringes reach minimum visibility. How far is the mirror 
moved? 

4. A certain spectral line which is a close doublet has a mean wave- 
length of 3440 angstroms, and a separation between the components of 
0.0063 angstrom. If the mirrors of a Fabry-Perot interferometer have a 
reflecting power of 85 per cent, what must be their separation to resolve 
the doublet? What resolving power is indicated? Assume the width 
of each component to be less than 0.002 angstrom. 

6. What is the resolving power of a Fabry-Perot interferometer in 
which the separation is 15 mm., for a reflecting power of 75 per cent? 
For a reflecting power of 90 per cent? (Assume X = 5000 angstroms.) 

6. What will be the effect on the resolving power of a Fabry-Perot 
interferometer if one of the plates has a reflecting power of 60 per cent 
and the other 80 per cent? Will it make any difference which plate has 
the higher reflecting power? 



CHAPTER XII 



DIFFRACTION 

In Sec. 9-9 it was shown that if a plane wave from a distant 
point S is incident on a slit of width a, the result will not be a 
sharply outlined single image of the slit but a series of images 
separated by regions of zero intensity, forming a diffraction pat- 
tern. The effect produced by diffraction is not to be confused 
with that obtained with an instrument fulfilling the conditions 
for interference proper. To be sure, the pattern of maxima and 
minima in diffraction is due to the reinforcement and cancellation 
of parts of wave fronts exactly as in interference, but by the 
principle of superposition it is shown that true interference may 
be obtained with no limitation whatever on the extent of the 
wave front. With certain types of interferometers, both true 
interference and diffraction are present. 

1. Fresnel and Fraunhofer Diffraction. Phenomena of this 
kind, i.e., those which owe their appearance to a limitation of 



* 
s 




FIG. 12-1. Fresnel diffraction at a slit. 

the wave front, are divided into two general classes. When the 
wave front from a source, not necessarily at an infinite distance, 
passes one or more obstacles and then proceeds directly to the 
point of observation without modification by lenses or mirrors, 
the resulting phenomenon is known as Fresnel diffraction. When 
the wave front incident upon the obstacles is plane, either from a 
distant source or by collimation, and the diffracted light is 
focused by a lens or mirror, or is observed at a distance infinitely 
far from the obstacle, the result is known as Fraunhofer diffrac- 

164 



SBC. 12-2] 



DIFFRACTION 



165 



tion. The difference between these types of diffraction may be 
further illustrated by a comparison of the forms of the diffracted 
wave fronts. In Fig. 12-1 light from a source S is intercepted 
by a slit so that only the portion AB is transmitted. With P 
as a center, strike an arc CD. The effect at P, due to Fresnel 
diffraction, is the result of the summation of all the disturb- 
ances which occur along CD at the same time. While it must 
not be supposed that a wave front actually occurs at CD, it is 
possible to define the surface thus represented as the diffracted 
wave front with reference to the point P. In Fraunhofer diffrac- 
tion both the real wave front incident on the obstacle and the 
diffracted wave front are plane, as shown in Fig. 12-2. 



* 
S 




B 




FIG. 12-2. Illustrating the existence of a real wave front CD in Fraunhofer 
diffraction. The existence of CD may be deduced from the Huygens principle. 

2. Fresnel Zones. While many of the important applications 
of diffraction are of the Fraunhofer class, the methods developed 
by Fresnel constitute a simple approach to the theory of diffrac- 
tion and will be considered first. From a consideration of the 
Huygens principle, Fresnel was led to the conclusion that light 
of a given wave-length from a point (Fig. 12-3) will produce 
the same illumination at P, no matter whether it passes directly 
from to P or is regarded as due to the summation of the effects 
at P of all the Huygens wavelets originating along W, a wave 
front proceeding originally from 0. He divided the wave front 
into many elements in the following manner: Draw lines PM\ t 
PM 2 , etc., in Fig. 12-3 such that 

PM i - PMo + ; PM 2 - PM l + ; etc. 



Then if the figure is rotated about OP, each such pair of lines will 
enclose a zone whose distance from P is X/2 smaller than the 



166 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 

one outside it. Because of the diminishing width of the zones 
due to increasing obliquity of the rays to 0, the amplitude of the 
disturbance at P due to light from outer zones is less than that 
due to inner ones. The total amplitude at P is the sum of a 
series of terms a\, a 2 , a 3 , etc., which alternate in sign because 
the disturbance from any zone is opposite in phase to the dis- 
turbance from adjoining zones, so that we may write 

S ai a 2 + a 3 o 4 + a n , 

in which each term of the sum is a little smaller than the one 
preceding it. It can be shown 1 that, taking into account the 

w 

V 






O 



I 

FIG. 12-8. 

smallness of the differences between the terms and the regularity 
of their change, the sum is 

e a\ . a n/ 

6 - 2 + T 

and when a n is very small, the effect is that of half the first zone. 
This brings the Huygens principle essentially into agreement 
with the rectilinear propagation of light, when the wave front 
is not limited by obstacles. 

3. The Zone Plate. If the light from a point source is passed 
through a circular aperture, the total effect of all the half-period 
elements passing through it to a point on the axis can be obtained. 
In Fig. 12-4 the circles are the boundaries of half -period elements 
whose distances to the point P differ by X/2. The areas of the 
zones enclosed by these circles are 





!See Schuster, "Theory of Optics," Chap. V. 



SEC. 12-3J 



D1WKAVT1UN 



ID/ 



7r(r 2 2 - n) = 7r(PM 2 2 - 



- J 



( + X) 2 




etc., where s is the distance from the zone plate to the image and r 
gives the radius of each circle, so that if X is small enough com- 
pared to the other dimensions, X 2 is negligible and the area of each 
zone is irsX. Consequently the consecutive zones will, because 
of the approximate equality of their obliquity, almost cancel 
each other. But if alternate zones are blocked out so that they 
transmit no light, the remaining ones will give an image at P. A 
series of transmitting zone apertures of this sort is called a zone 
plate. If s is required to be 100 cm., for a wave-length of 0.00005 
cm., the area of each zone will be about 0.0157 sq. cm. 

etc. 
M 4 

;,-M 3 







,M O 



o 



FIG. 12-4. 

The radii of the zone boundaries are r\ = -\Ax, r 2 = \/2sX, 
etc., hence a zone plate may be constructed by first drawing on a 
large sheet concentric circles whose radii are proportional to 
the square roots of the consecutive integers 1, 2, 3, etc., and then 
blackening alternate zones. This drawing can be copied photo- 
graphically in any desired size. The negative, or positive, thus 
obtained may be used to produce an image of a distant object, 
such as the sun, at a distance s from the plate. Thus the smaller 
the reproduction the shorter will be its " focal length." The 
intensity of the image produced with a zone plate will be greater 
if alternate zones are not blocked out but are left transmitting, 
with a phase difference of one half period introduced between 
them and adjacent zones. This can be done with some degree 
of success by covering a glass plate with a thin coating of wax 
which is then scraped away in the annular area corresponding to 
alternate zones. The plate is then etched slightly with dilute 



168 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 



hydrofluoric acid to the proper depth. Obviously this method is 
limited to the few zones which are of sufficient width to permit 
the treatment. 1 
4. Diffraction by a Circular Obstacle. If a circular obstacle is 



hiternosed between a Doint source of light and the observer, all 







JPIO. iz-o. Lsmracnon 01 iigm oy a circular oostacie. \a) snows tne origin 
image point at the center of the shadow. (6) the same as (a) but with distances 
so chosen that distinct circular fringes also appear in the shadow. The fringes 
outside the shadow are analogous to those in Fig. 12-7. (c) the same as (a) and (6) 
but with a circular obstacle 20 cm. in diameter held in the hand, and the light 
condensed by a lens so as to make the virtual distance from source to screen 
equal to 7 km. (d) was taken with the monogram of the letters I and U as a 
source instead of a point source, (a), (6), and (c) copied from Arkadiew: Physi- 
kalische Zeitschrift, 14, 832, 1913. (d) copied from M. E. Hufford, Physical 
Review, 3, 241, 1914. 

the light due to a number of central zones will be obstructed. In 
this case, by summing up the amplitudes due to the remaining 
zones in_the way^joutlined Jn.Sec..jL&8i it will be found that the 
resulting disturbance is that due to one-half the first zone to 
pass the edge of the obstacle. If the obstacle is not too large, 
there should be on the axis at the center of the shadow of the 
obstacle an image of the point source of practically the same 

1 Some interesting details on the construction of zone plates are given in 
Wood, "Physical Optics," Macmillan. The subject is treated more fully 
in t.ho edition of 191 1 than in the later edition of 1934. 



SEC. 12-3J 

intensity as if the obstacle were not there. This result was 
deduced by Poisson, who considered it an argument against the 
validity of FresnePs theory. Thereupon, Arago performed the 
experiment, and showed that the image actually exists as pre- 
dicted. The effect is illustrated in Fig. 12-5. 

5. Cylindrical Wave Front. When the source is long and 
narrow, as in the case of a hot filament or an illuminated slit, it 
is convenient to consider the zones to be not concentric rings 
but rather strips parallel to the source on a cylindrical wave 
front. Let (Fig. 12-5A) represent such a source, and W a 
cylindrical wave front whose axis is perpendicular to the page at 
0. Then as before on either side of M points Afi, M 2 , etc., may 
be chosen such that the distances of the successive zones Ma 




* 
o 





i/ M 




a 


Af' 


b 




'* 




"-A/J 




etc. 





Fio. 1 

etc., from a point P differ by X/2. It is evident from Fig. 
12-5 A that the adjacent strips, which may be called half-period 
zones, will differ in area rapidly at first, and then more slowly as 
strips of higher number are considered. The outer ones will be 
practically equal in area and their effects at P will cancel each 
other, so that the amplitude at P will be due to only a few strips 
about the point M o, which is known as the pole of the wave front 
with respect to P. 

*%&. Diffraction at a Straight Edge. Consider light from a line 
source perpendicular to the page in Fig. 12-6, passing a straight- 
cdged obstacle B to a screen. It is required to find the illumina- 
tion at any point on the screen. Let us first consider a point P' 
above P on the screen, well outside the geometrical shadow of 
the obstacle. A straight line drawn from to P' intersects the 
wave front at the point B', which is thus the pole of the wave 
front with respect to P', and by the arguments of the preceding 
section the amplitude at P' will be due only to the half-period 



170 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 



zones in the neighborhood of B' ' . If P' is sufficiently far away 
from P so that the obstacle imposes no limit on the elements 
effectively contributing to the amplitude at P', full illumination 
will exist, but if P' is a point on the screen near enough to P so 
that the effective half-period zones about the pole B' are partly 
obstructed, the amplitude at P' will suffer a modification depend- 
ing, in the final analysis, upon whether or not there is an odd 
or an even number of zones between B and B'. If the number is 
even, their contribution to the amplitude at P 1 will be a minimum, 




(a) (b) 

Fia. 12-6. Diffraction at a straight edge. The source is a slit at O perpendicular 

to the plane of the paper. 

since alternate ones are opposite in phase; if the number is odd, 
the amplitude from them to P' will be a maximum, since all but 
one tend to cancel each other. 

As points farther and farther into the region below P arc con- 
sidered, the farther will their poles lie along the wave front 
below the edge of the obstacle, and the smaller will be the number 
of zones contributing to the amplitude on the screen. Hence 
the intensity below P will fall off gradually to zero. 

For a maximum at P', BP 1 - B'P' = (2n + l)X/2, and for a 
minimum, BP' B'P' = 2nX/2, where n is zero or any integer. 
Now if OB = a, BP = 6, and PP' = x, 



BP' = 




, 6 2 



approximately 6 




OP' = v (a + 6) 2 + x z = approximately a + 6 + 



TV 
o) 



SBC. 12-6] 

Hence for a maximum, 



DIFFRACTION 



171 



(2n 



26 2(a + 6) 



or, 



X = \ - a ^ W 
* ~ \ - a - ^ n ^ 



a 



a mm > mum > 



where n is zero or an intecror. Thus the r1iffra.rt.inn nn.t.fnm 




FIG. 12-7. Photograph of diffraction at a straight edge. 

be a series of maxima and minima as shown in Fig. 12-66. A 
photograph of the pattern is shown in Fig. 12-7. 

6. The Cornu Spiral. The explanation of the intensity dis- 
tribution observed in certain diffraction patterns of the Fresnel 
type which has been presented in the preceding sections is 
quite elementary, and is limited in its applicability. It serves, 
however, to furnish an introduction to a much more elegant 
method of representing the disturbance at any point in a diffrac- 



172 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 



tion pattern. This method, due to Cornu, makes use of the 
expedient of dividing the wave front into elements, which are 
thereupon summed up by an extension of the vector-polygon 
method introduced in Sec. 9-5. 

In Sec. 12-5, in order to represent the amplitude at a point P, 
the wave front was considered to be divided into half-period 
zones MoMi, M\M^ etc., as in Fig. 12-5A, on either side of the 
pole MQ of the point P. The zone M M i was constructed by 
choosing a point M \ such that the distance M\P is X/2 greater 
than the distance MoP; similarly M Z P is X/2 greater than M\P, 







(d) 



(e) 



FIG. 12-8. Application of the vector polygon method, (a) The sum of two 
half-period zones of different amplitudes and opposite phase; (6) the sum of 
two half-period zones of the same amplitude and same phase; (c) and (d) vector 
polygons of eighth-period dements which together give the same as (a); (e) 
vector polygon of eighth-period elements giving the same as (6) . 

and so on. On the other side of the point M , the corresponding 
zones are Af MY, Mi'MJ, etc., the entire system of zones thus 
being symmetrical about the point Af . If it is desired to 
represent graphically by a vector polygon the amplitude at P due 
to the zones M M i and JlfiAf 2 , the polygon will consist of a pair 
of vectors as indicated in Fig. 12-8a. These are parallel and in 
opposite directions because the phase difference between them 
is TT; the vectors are not equal in length because the area of 
zone AfoMi is greater than that of zone M ]M 2 . Similarly in 
Fig. 12-86 is the vector polygon summing the amplitudes due 
to the two zones Afi'Afo and AfoAfi on either side of Mo, the 



SEC. 12-6] 



DIFFKACTION 



173 



resultant amplitude being given by the vector Mi Mi. It is 
evident that by this method it will not be possible to represent 
the amplitude due to any part of the wave front unless it con- 
tains a whole number of half-period zones. Suppose, however, 
each zone is divided into k smaller elements, such that adjacent 
ones differ in phase, not by fr, but by ir/k. For instance, if k is 4, 
the phase difference between successive elements will be x/4, and 
the vector summation of the amplitude at P due to all the ele- 
ments in the zone M Mi will be that shown in Fig. 12-8c, that 




Fio. 12-9. The Cornu spiral. The convolutions close up spirally to J and J'. 
The distance along the curve from Mk-i to Ma corresponds to a half convolution, 
representing a half-period zone. The distance from to Mk-i represents two 
half-period zones opposite in phase. 



for MiM* will be that in Fig. 12-8rf, and for Mi'M + M Q Mi 
that in Fig. 12-8e. Also, it is now possible to sum up the ampli- 
tude over fourths of zones. If the number of these small 
elements in each zone is made very large, each vector will be corre- 
spondingly small, and the succession of vectors representing the 
elements will be a smooth curve. The curve representing all 
the elements on both sides of Mo for an unobstructed wave is known 
as the Cornu spiral, shown in Fig. 12-9. The entire curve is 
not drawn, but the two arms are terminated in convolutions, of 
which only about two are shown, which become smaller and 
smaller and more nearly circular until they are finally asymptdtic 
to circles of zero radius at J and J'. The straight line joining 
J and J' and passing through the origin represents the entire 



174 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 



wave when it is unlimited by obstacles. Referring once more 
to Fig. 12-5A, in this case the point P will receive full illumination. 
Suppose, however, an obstacle is brought gradually in front of 
the source from one side so that as it advances, it cuts off more 
and more of the wave front. In doing so it will cut off successive 
half-period zones, each of which is represented in the Cornu 
spiral by a half convolution such as MkMk-i (Fig. 12-9), arid 
the vector representing the summation will no longer be the line 
joining J' and J, but will be a line joining J' and a point which 
moves along the spiral from J toward 0. The corresponding 
illumination from the source will alternate between maximum 
and minimum. When half the wave front has been cut off, i.e., 
when the obstacle is at the pole of the point P (Fig. 12-5 A), the 




(b) 

Fio. 12-10. 



(O 



total amplitude will be represented 011 the Cornu spiral by the 
vector J'O. 

Only two of the many graphical solutions of problems in 
diffraction will be mentioned; the case of the straight edge and 
the case of the single slit. 

a. The Straight Edge. Actually this case has just been 
described in considering an obstacle gradually brought in from 
the side 86 as to obscure more and more of the wave front, except 
that the alternations of intensity were considered as taking place 
at a single point on a screen as the obstacle advanced. If, 
instead, point P' in Fig. 12-6 is considered to move along the 
screen toward P, then the intensity at the moving point will 
alternate. On the Cornu spiral (Fig. 12-10a), a few alternations 
in amplitude will be in the order J'Af 3 for a maximum, J'M Z for 
a minimum, J'M\ for a maximum, and thenceforth the amplitude 
will diminish until J'O (Fig. 12-106) represents the amplitude 
at the geometrical edge of the shadow on the straight line from 



SEC. 12-6] 



DIFFRACTION 



175 



the source past the edge of the obstacle. Thenceforth the ampli- 
tude vector J'M (Fig. 12-10c), will reduce gradually in length 
to zero, as M moves along the curve, representing the gradually 
diminishing illumination on the screen in the shadow. 




FIG. 12-11. 

6. The Single Slit. Consider a slit (Fig. 12-11) so narrow 
that only the two central zones, one on each side of the pole of P, 
contribute light to P. The amplitude at P will then be given 
by the vector Mi Mi (Fig. 12-12a). As points above P are 
considered, the amplitude will be given by a series of vectors 




(a) (b) 

Fia. 12-12. Amplitudes in Fresnel diffraction at a slit. 

such as Mz'Mz until when P' is reached, for which the^edge of 
the slit is the pole, the amplitude is given by ON. Hence, when 
the slit width is only two zones, the center of the pattern is a maxi- 
mum. Suppose, however, the slit is as wide as four zones. Then 
the amplitude at P (Fig. 12-11), is given by N'N (Fig. 12-126), 
and for points just above P by M\M\ y which is longer than 
N'N, so that in this case the intensity at the middle of the 
pattern is a minimum. For cases where the aperture is smaller 
than a single zone, the vector joining the two points in the spiral 
is so short that it can be moved along the spiral in either direc- 



176 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 



tion without alternations in length, but with its greatest length 
when it extends equal distances on each side of 0, indicating that 
th^re will be a maximum at the middle of the pattern. 1 

7. Fraunhofer and Fresnel Diffraction Compared. While it 
is possible to make use of the graphical methods outlined in the 
preceding sections to describe diffraction effects of the Fraun- 
hofer class, the difficulties involved in a mathematical analysis 
are far less than in the case of Fresnel diffraction. This is 
because in Fraunhofer effects the diffracted wave is plane, mak- 
ing possible a fairly simple method of summing up analytically 



\ 



\ 



* 

s 





FIG. 12-13o. Fresnel diffraction through two parallel slits. While inter- 
ference fringes will appear across the entire field, their visibility will he greatest 
at the middle point. 

the disturbances reaching any point, while in the Fresnel case the 
diffracted wave front is not plane. 

The experimental advantages of Fraunhofer over Fresnel 
diffraction are really found in those cases where the effects 
to be observed are produced by the interference between two or 
more beams. The diffraction is present for the reason that these 
particular interference phenomena are without exception pro- 
duced by apparatus belonging in class A (see Sec. 10-4) in which 
a division of the wave front is made. In all apparatus such as 
the grating, the echelon, and the Michelson stellar interferometer, 
the elements recombined are relatively narrow sections of the 
wave front from a slit source or a source of small size, and the 
recombination necessary for interference is effected by focusing. 
A simple experiment will serve to illustrate the point. Figures 

1 For a more complete discussion of the Cornu spiral, see Meyer, "The 
Diffraction of Light, X-rays, and Material Particles," University of Chicago 
Press, 1934. There is also a comprehensive treatment in Preston, "Theory 
of Light," 4th ed., Macmillan. See also Appendix VIII of this text. 



SBC. 12-7] 



DIFFRACTION 



177 



.3 




'a 



g 
- 

g 



feO 
oo 



gcj 

r fti 
S 

= 5 



us 

Jl 

**. 

^ O- 



33 

a * 



CSI 



178 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 



12-13o and 6 illustrate Fresnel diffraction through two slits. 
Obviously the effect due to the superposition of the two diffracted 
beams occurs in a region of poor illumination. It is true that by 
decreasing the width of each slit the regions of greatest intensity 
could be made to spread out until they overlapped, but this would 
be at the expense of illumination and little would be gained. If a 
lens system is used, as illustrated in Figs. 12-14a and 6, the most 
intense portions of the diffraction patterns may be superposed 
at the focus, and the interference fringes will be observed in the 







FIG. 12-14a. Fraunhofer diffraction through two parallel slits. The visibility 
of the fringe system is highest in the region of maximum intensity. 

brightest part of the field. This is the advantage which the 
Rayleigh refractometer has over Young's experimental apparatus 
(see Sees. 10-6 and 10-8)* The two are analogous, but the 
former uses Fraunhofer and the latter Fresnel diffraction. In 
Young's apparatus the two slits from which the interfering 
pencils of light come are so close together that it cannot be used 
for comparisons of optical path. 
78. Fraunhofer Diffraction by a Single Slit. 1 This is illustrated 

1 This is the first of a series of treatments, each of which gives the intensity 
distribution in a pattern due to diffraction. In many texts it is customary 
to adopt a standard method of derivation which is thereafter applied to each 
case in turn. The author feels that in an intermediate course the methods 
of analysis are often as valuable an acquisition for the student as an under- 
standing of the phenomena themselves. Moreover, a variety of treatment 
often enhances the understanding of the entire field. For this reason 
different approaches have been made as often as feasible to the cases of 
diffraction treated in this chapter. 



SEC. 12-8J 



DIFFRACTION 



179 




180 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 

by Fig. 12-2, but to assist in the derivation, a diagram showing 
more detail is desirable. In Fig. 12-15 a plane wave train of 
wave-length X is incident upon a slit at an angle i to the normal 
to the slit. All parts of the incident wave are in the same phase 
of disturbance and can be represented by the expression 



, 
s c sin 



r 



Each part of the wave front passing through an element dx of 
the slit will be out of phase with that passing through the middle 
by an amount 2ir8/\, where 8 is the total difference of path 61 -f 5 2 
between that part of the wave front which traverses the center 
of the slit and that which traverses the element dx in passing to 
the diffracted wave front. By the geometry of the figure, it 




FIG. 12-15. 

follows that 6 = z(sin i sin 0), where x is the distance from 
the middle of the slit to the element dx and 8 is the angle of 
diffraction. By convention the positive sign is used for sin 9 
when the angle is as represented in the figure, and the negative 
sign for diffraction. to the other side of the normal to the slit. 
The disturbance in the elementary pencil of light from dx will 
thus after diffraction be of the form 

, , . n \ t x (sin i + sin 0)~|, , . 

ds' = c sin 2ir\ -~ =- \dx. (12-1) 

In order to obtain the entire disturbance after diffraction, the 
function of x given in eq. 12-1 may be integrated between the 

limits and +: 



SBC. 12-8] 



DIFFRACTION 



181 



ff = c f + /2 S in fcf i - f 

J-a/2 L-* A . 



(12-2) 



in which for convenience ^ is substituted for the quantity 
sin i -f sin 0. Expanding the sine function in eq. 12-2 



j /+o/2 

' = c sin 2ir7p I cos 

* J-a/2 



c cos ? 



sn ir- 

A 



(12-3) 

in which the second term on the right-hand side, being an even 
function, is equal to zero. Hence 



c sin 2ir~ 



sm 



-o/2 



Sin IT-r- 

ac - ~ sin 
<f>a 



(12-4) 



Thus the disturbance in the diffracted wave is of the same form 
as in the incident wave but has instead of a constant amplitude c, 
the amplitude 

ac sin 7r(g?a/X) 



which depends upon the width of the opening, the wave-length, 
and the angles of diffraction and incidence. The intensity of 
the resulting diffraction pattern, after the light has been brought 
to a focus by a lens, is proportional to the square of the amplitude. 
Considering the proportionality factor to be unity, we have from 
eq. 12-4, 

a 2 c 2 sin 2 

7 = 



For simplicity, we may put this equal to (sin 2 u)/u 2 and proceed 
to analyze the intensity pattern as follows : 

a. When u = 0, (sin u)/u is an indeterminate quantity which 
evaluated gives unity. This corresponds to <p 0, the middle 
of the diffraction pattern, and to i, or, in other words, to a 
position directly opposite the source. 



182 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 



b. When u = mw, for m = 1, 2, 3, etc., the intensity will be 
zero, representing a series of equidistant minima on either side 
of the central maximum given by case a. It should be noticed 
that the distance between the two minima corresponding to 
m = 4-1 and m 1 is twice the distance between any other 
two adjacent minima. 

c. Between the minima will exist a series of maxima whose 
positions cannot be found by inspection. To locate them we may 
put the first derivative of the intensity with respect to u equal 
to zero. 

dl _ 2 sin u 
du u" 



-(u cos u sin u) = 0. 



(12-6) 



The first factor on the right-hand side gives the cases (a) and (b) 
already discussed. The second factor put equal to zero can be 




FIG. 12-10. 

written tan u = u. In order to find the values of u satisfying 
this equation, we may find the values which satisfy simultane- 
ously the equations y = tan u and y = u. These graphs are 
shown in Fig. 12-16. The intersections of the solid lines are the 
required values of u. The dashed lines parallel to the ^/-axis 
give the values of u for which the angle is 7r/2, 3ir/2, 57T/2, etc., 
and the dotted lines the values corresponding to the positions 
of maxima. It is apparent that the maxima do not lie midway 



SEC. 12-8] 



DIFFRACTION 



183 



between the minima but are displaced somewhat toward the 
middle of the pattern, the displacement being greatest for 
the maximum of lowest order. The values of u for the central 
maximum and the first six maxima on either side are given below, 
together with the relative intensities, the intensity for the 
central maximum being taken as unity. 



Order of 
maximum 


u 


Intensity 








1.0000 


1 


1 . 4307r 


0.0469 


2 


2.4597T 


0.0168 


3 


3.471ir 


0.0083 


4 


4.4777T 


0.0050 


5 


5.4827T 


0.0034 


6 


6.484T 


0.0024 



An examination of eq. 12-5 discloses that if the slit is made 
narrower the entire diffraction pattern will broaden. Since the 




= -J/7 ~2/7 -ii +ii 

FIG. 12-17. Graph of intensity distribution in the diffraction pattern of u 

single slit. 

minima occur where ira<(>/\ = mir, a smaller value of a corre- 
sponds, for any given value of m, to a larger value of <p, and hence 
of sin 0. Increasing the angle of incidence will also result in a 
broadening of the pattern, since it will diminish the effective 
aperture, which is a cos i. An increase in X will also correspond 
to an increase in v for both the maxima and the minima. Henco 
if light of more than one wave-length be incident on the slit each 
maximum except the central one will consist of a spectrum whose 



184 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 

violet end will be closer to the middle of the pattern than the red 
and whose width will be proportional to the order of the maxi- 
mum. A graph of the diffraction pattern of a single-slit opening 
with monochromatic light is shown in Fig. 12-17. A photograph 
of the diffraction pattern obtained with one slit is shown in 
Fig. 12-26. 

'9. Two Equal Slits. If two parallel slits are used, the resulting 
disturbance after diffraction can be found analytically by the 
same procedure as followed in finding that for a single slit. For 
simplicity consider the two slits to be of equal width a, and 
separated by an opaque space 6. Each part of the wave front 
passing through an element dx in either slit will, as before, be 
out of phase with that passing through the middle of the first 
slit by an amount 27r5/X. Consequently eq. 12-1 gives the 
disturbance ds' in each element of the diffracted wave, and to 
find the disturbance in the entire diffracted wave front, the 
integration over the entire wave front passing through two slits 
may be performed. The total disturbance will be 

, f. r . N f +o/2 , /+3a/2+6 

S r (two slits) = I ds' + I ds', 

V ' J-a/2 J+a/2+b ' 

since the distances of the boundaries of the two slits are the 
limits of integration given. The result is 



, n sin (ira<f>/\) TT(O + b)<p . 2irt /10 -, 
S = 2ac - -^ - cos r sin -7=-; (12-7) 

r \ / 



and the intensity, given by the square of the amplitude factor, 
is then 



As would be expected, a comparison of eqs. 12-5 and 12-8 shows 
that the intensity of the maximum for two equal openings is four 
times the intensity for a single opening of the same width, the 
amplitude being twice as great. Also, except for the factor 4, the 
expression for the intensity is the same as that for a single slit 



multiplied by the factor cos 2 w -^-- , which varies between 

A 

unity and zero for positive and negative values of <p, and hence 
for positive and negative values of 0. Two important features 



SBC. 12-9] 



DIFFRACTION 



185 



in the intensity pattern for two slits then follow: (1) The 
distribution of intensity due to a slit of width a, which will be a 
diffraction pattern like that shown in Fig. 12-17; and (2) super- 
posed upon this a series of maxima and minima whose spacing 
is determined by the values of a and 6, which will be a series of 
interference fringes in which the maxima are limited in intensity 
by the diffraction pattern. To show the result graphically we 

may first construct the graph of 4o 2 c 2 , /x ( 9 , and under the 

2 




I TT- 



7i 2ir 

FIG. 12-18. Graph of the intensity distribution in the diffraction and inter- 
ference pattern due to two equal slits, for which 6 = 3a. 

curve thus obtained, draw the graph of cos 2 [ir(a + b)<p/\]. By 
inspection of the latter function we see that there will be a series 
of minima when ir(a + b)<f>/\ = (2m 4- 1)^/2, and a series of 
maxima when ir(a 4- b)v/\ = mir, where m = 0, 1, 2, 3, etc., or 



(12-9) 



, . . x (2m -f , 

v?(mmima) = -^ ,> = sin i H- sin 0, 

(L -f- o) 



**V(maxima) = 



2mX 



2(o + 6) 



T = sin i 4- sin 6. 



Hence the maxima will be evenly spaced and midway between 
the minima. The intensity of a maximum for a particular value 

flf m will in all cases be limited by the value of , /x ( 9 > but it 
"* * (ira<t>/\) 2 

pend also upon the relation between b and a. If b is equal 

to an integer times a, there will be a value of m corresponding to 
a maximum of the two slit interference pattern which will be 



186 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 

located at the minimum of the diffraction pattern. The graph 
shown in Fig. 12-18 is for the case where b = 3a. 

The pattern obtained with two or more openings is an inter- 
ference pattern since it is due to the superposition of separate 
beams, originally from the same source, in such a manner that a 
regular distribution of maxima and minima of intensity is the 
result. 

10. Limit of Resolution. According to Rayleigh's criterion, 
the limit of resolution with a single opening is reached when the 
two objects are such a distance apart that the central maximum 
of the diffraction pattern of one object coincides with the first 
minimum of the diffraction pattern of the other. We may 
consider the two objects to be two parallel incandescent fila- 
ments, or two slits close together and illuminated, or, in fact, 
any two sources of light parallel to the diffracting slit. In Sec. 
12-8 it was shown that the first minimum of the diffraction 
pattern of a single object occurs when u = ira<f>/\ = TT, or when 
V X/a. But <p = sin i + sin 0, so that for normal incidence, 
and provided <p is not too large (i.e., provided a is not too small), 

<p = = - (12-10) 

f L 

If, however, the intensity pattern is due to two slits, it follows 
from eqs. 12-9 that the limit of resolution is reached when 



If a has the same value in both eqs. 12-10 and 12-11, it follows 
that the angular separation of two objects which are just resolved 
is less than half that for a single slit, the exact ratio between the 
two angular separations depending on the value of b. The 
larger b is, the smaller will be the angle for the limit of resolution 
with two slits and the greater will be the resolving power. If, 
moreover, the a in eq. 12-11 is a', much smaller than a in eq. 
12-10, and the value of a' + b is the same as that of a, the 
resolving power of the two slits will be twice that of the single 
opening. This condition is approximated in the case of the 
stellar interferometer which will be discussed later. In the 
simplest form of this instrument, the central part of a lens of 
width a is covered up, permitting the light to pass only through 



SEC. 12-11] 



DIFFRACTION 



187 



two narrow openings at the edge, whose separation is a' -f 6, 
where a' is the width of each opening and b is the width of the 
cover. In this case, a' + b is approximately equal to a. 

There is a difference, however, in what is observed with one 
and with two slits. If diffraction images of two objects are 
obtained with a single slit, at the limit of resolution the graphical 
representation of the result is that given in Fig. 12-19o. Here the 
central images are so much greater in intensity than the others 
that in most cases they are the principal observable features, 
especially if the slit is wide. If two slits are used, at the limit of 
resolution the graphical representation is that given in Fig. 

AB 





i i 
f\ 



(a) 



O 

(b) 




Fig. 12-19. The difference between superposition at the Rayleigh limit in the 
case of (a) diffraction by a single slit, and (6) interference by two slits. 

12-196, where for each image there are several maxima not 
differing much in intensity. The resulting effect of the super- 
position of the two interference patterns with the separation 
shown in the figure is a disappearance of the fringes. If the 
superposition is not at the limit of resolution, some alternations 
of intensity will be visible, the maximum visibility of the fringes 
being when the dotted curve is exactly superposed on the one 
represented by the solid line, but with a separation between 
points A and B equal to the distance between two adjacent 
maxima. The test for limit of resolution is then determined 
by the degree of visibility of the fringes. This principle is made 
use of in Michelson's stellar interferometer. 

11. The Stellar Interferometer. It now remains to show 
that the angular separation between the diffraction (or interfer- 
ence) maxima is the angular separation between the objects. 



188 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 



This may be done with the aid of Fig. 12-20. Points 0\ and 2 
represent two very distant objects, and the straight lines con- 
necting them to /i and 1 2 are the chief rays of the rays collected 
by the lens which focuses the light at the image plane. The 
distance between I\ and /a in the case of one opening is, for the 
limit of resolution, the distance between the principal maximum 
of /i and the first minimum in its diffraction pattern. 

In the case of two slits, the situation requires a more com- 
plicated diagram. At the limit of resolution, given by the 
disappearance of the interference fringes, the angle 6 is, in this 
case, the angle between the two central maxima of the inter- 




(a) 




Middle 
of I t 



X5^ Middle 



(b) 



FIG. 12-20. (a) Shows the relation between the angular separations of the 
two objects and the two images with one slit. The lens has been omitted, (b) 
Shows the corresponding case for two slits. The distance a + b is from center 
to center of the slits and the distance e is from the upper slit to the inclined dotted 
line. There is no relation between the scales used in the two diagrams. 

ference pattern, these maxima being indicated by A and B in 
Fig. 12-196. The angle 0, subtended at the slit plane by the dis- 
tance between the maxima A and B, is the same as that subtended 
at the slit plane by the distance between the two point objects. 
From Fig. 12-206 

Bin B = = -4-r> 
a 4- 6 

but by eq. 12-11, 

A 



- 



2(o + 6)' 



when the limit of resolution is reached. These two values of 6 
will be the same when e = A/2, but since this is true when the 
fringes disappear, it follows that for a separation of the two slits 



SBC. 12-11] DIFFRACTION 189 

such that the visibility of the fringes is a minimum as shown, the 
separation s of the two objects is given by 



I I i 



D 2(a + b) 

Similarly, the disappearance of the fringes, or, rather, the 
adjustment of the separation between the slits so that the visibil- 
ity of the fringes is a minimum, may be made 
use of to measure the diameters of stellar objects 
whose distances are known. The application 
of the principle in this case may by illustrated 
by considering the case of a distant slit con- 
sidered as a source. Let 1, 2, 3, 4, 5, ... n 
(Fig. 12-21) represent elements in the plane of 
the source parallel to its sides. For each of 
these elements there will be, because of the 
double slit, a pattern of equidistant maxima . Fia ' . 12 - 21 -~ For 

. . two point sources 

and minima. Patterns from elements 1, 2, the zones i, a, 3, 

3 . . . n will be superposed as shown graphi- : n each gl an 

!~ . r i interference pattern 

cally in Fig. 12-22. It is evident that unless the of the type iiius- 
angular separation of elements 1 and n at the * rate< * m F J8- 12 :?2 

by the intensity 

plane of the double slit is B = A/2 (a -f- 6), curves correspond- 
alternations of intensity will still be observed, in ly numbered. 
i.e., the visibility of the fringes will not be a minimum. If, how- 
ever, the angular separation of elements 1 and n is A/2(a -f- 6), 
the visibility will be a minimum, and zero in case the intensity is 
uniform across the source. 

An analytic treatment of the visibility of the fringes may be 
based on further consideration of eq. 12-8. In this equation, 
the quantity (a + b)<p is the total difference of path for light 
diffracted at a particular angle 6 through the two openings, and 
(a + 6)v>A is consequently the number of wave-lengths n in 
that difference of path. Also, since in practice the angles i and 6 
are very small, we may substitute the angles for their sines, so 
that <p ** i + 0. Thus for an element of the source of width di t 
considering that for small angles sin (ira^/A) = ira<p/\, we may 
write eq. 12-8 in the form 

- 4B 2 cos 2 *n(i + 0) (12-12) 



where B is the amplitude for a particular point in the diffraction 



190 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 



pattern from either opening, and n is the number of wave- 
lengths difference of path. The total intensity will be 

/ = f +a/2 4B* cos 2 irn(6 + i)di, (12-13) 

J a/2 

where a is the angular width of the source as seen from the posi- 
tion of the double slit. Since the double slit is in front of the 
objective of a telescope, a is the angular width subtended at the 
telescope. If the intensity is uniform over the source, the value 




FIG. 12-22. Each of the curves 1, 2, 3, ... n indicates a maximum of an 
interference pattern from a line element in the source, due to two slits. The 
dotted curve at the top of the picture indicates that, unless the two slits have a 
suitable separation, there will be maxima and minima of intensity in the com- 
posite pattern. 

of B will be the same for all elements di, so that eq. 12-13 may be 
written 



I = 2B 2 F C +a/2 di + f + " /2 cos 2irnO cos 2irni di 

LJ-o/2 J-a/2 



I 

J 



a 



-/2 



sn 



sn 



di 

. J 



Putting 



2 2 / sin 2irni di = S,) 
there results 

/ = p -f c cos 2irnd S sin 2irn6. 

The condition for maxima and minima of intensity is 

rf/ 
da 



(12-14) 



(12-15) 



sn 



cos 2ira0) = 0, 



SEC. 12-11] DIFFRACTION 191 

hence the intensity of the pattern of interference fringes varies 
between 



= P + VC 2 + 

and 



/nun = P - VC' 2 + S 2 

and the visibility of the fringes is given by 



V = {=-=4=5 . -- (12-16) 

* max l" * min 



If the source is symmetrically placed with respect to the axis of 
the telescope, S is an even function and becomes zero, whereupon 



( } 



-/2 

Thus the visibility is independent of B, which relates to a particu- 
lar place on the interference pattern under observation, i.e., V 
is a constant across the pattern, provided, of course, monochro- 
matic light is used. Since n = (a -f 6)/X and a = w/D, where 
w is the width of the source and D is its distance from the 
telescope, 

a -f b w 

sm "~~lT ' X 

V = -f - -- (12-18) 

a + b w ^ ' 

*~~D~ ' X 

From this it is evident that the visibility will be zero, i.e., the 
interference fringes will disappear, when j? is equal to 

L) A 

any integer m. That is, the width w of the source is given by 

wDX 



W = 



- j Y> 

a -f b 



where m = 1, 2, 3, etc. 

The first-order disappearance of the fringes will be for m = 1, 

or when 



w - i (12-19) 



192 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 

where d is the separation of the two openings in the double slit. 
The disappearances of higher order will in general be more 
difficult to observe, since they correspond to larger values of 6, 
the separation between the two slits, for which the fringes become 
very narrow. 

Should the focal plane of the eyepiece not be exactly at the 
focal plane of the objective of the telescope, the fringes will 
still be visible, provided the relation expressed in eq. 12-19 does 
not hold. In any case there is a separate diffraction pattern 
for each of the slits in front of the objective, and these two pat- 
terns will overlap in some part of their extent. The only effect 
of absence of correct focusing of the telescope eyepiece will be 
that the fringes may not be observed in the most intense portion 
of the field of view; they will in any case be present. Moreover, 
since the width of the two openings is inversely proportional to 
the widths of the resulting diffraction maxima, the fringes will 
still be seen (provided eq. 12-19 does not hold), even if each of 
the fringes has appreciable width. If white light is used, all 
of the fringes except the central one will be slightly colored at 
the edges, owing to the small amount of dispersion present. 
Since this dispersion is quite small, the disappearance of the 
fringes, or, at least, the reduction of the visibility to a minimum, 
may still be observed. In this case, the value of A in eq. 12-19 
will depend on the sensitivity of the eye to color. For most 
eyes the wave-length for maximum sensitivity is approximately 
5700 angstroms. 

If the distance between two illuminated objects is to be 
measured, their separation is given by 



- - (12-20) 



a result which may be derived by the preceding analysis. 

If the source is a circular disk of uniform luminosity, a series of 
strip elements on its surface will decrease in height as the edge 
of the disk is approached. For this reason the angle subtended 
at the telescope by the disk*must be somewhat larger than the 
angle for the disappearance of the fringes. Theory shows that 
in this case the diameter of the disk is given by 

(,2-2!) 



SBC. 12-11] 



DIFFRACTION 



193 



Similarly, in case the separation s of two stars is to be meas- 
ured, it is given by 

0.61XD 



a 



(12-22) 



where s is the distance between the centers of the stars. In 
eqs. 12-20, 12-21, and 12-22, a is the separation between the two 
slits. 

Observations by this method of the diameters and separations 
of celestial objects are usually made with a telescope whose 




B' 

A 1 



Fia. 12-23. The dotted circle represents the aperture of an objective; the 
two heavily shaded portions represent that part of the objective through which 
light passes. 

central portion is covered by a shield, so that there is used only 

light which passes through two narrow slots whose distance apart 

can be changed. From the four preceding equations, it will bo 

seen that the angular diameter of an object is proportional to 

w/D, and the angular separation of two 

ob j ects to s/D. Hence the smaller these 

quantities are, the larger must be the 

value of a, the linear distance between 

the slots, so that the measurement of A 

diameters or separations of very distant 

celestial objects, or those with relatively 

small dimensions, would be impossible 

except with a telescope objective of enormous size. For such 

objects, instead of having two movable slots over the objective 

as illustrated in Fig. 12-23, an arrangement of total reflection 

prisms is mounted on a crossarm placed in front of the objective, 

as illustrated in Fig. 12-24. While the prisms B and B' are 



A' 



FIG. 12-24. 



194 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 

primarily for the purpose of reflecting the two interfering beams 
into the objective, it follows from the third condition for inter- 
ference given in Sec. 10-2 that with this arrangement the fringes 
are sensibly wider and hence more easily observed than if the 
angle between the beams were larger. 

12. Many Slits. The Diffraction Grating. If the diffraction 
is by more than two equidistant slits of equal width, the equa- 
tion for the disturbance after diffraction may be obtained by 

integrating 12-1 between successive limits ~ to -f^' o ~^~ ^ * 

o (\ K y? 

-o" + &> ~o~ + 2 & to "o" + 26, etc. This procedure is so long and 

2i i t 

involved that it is likely to mask the significance of the final 
result. Therefore a more descriptive method of accounting for 
the resulting diffraction pattern will be used. 

In the case of two slits it was shown that interference maxima 
will occur at values of <f> for which the elements of disturbance 
from corresponding points in the two slits have path differences 
of mX, where m = 0, 1, 2, 3, etc., i.e., phase differences of 2mir. 
Correspondingly there will be minima for values of <f> which the 
difference of path is (2m + l)X/2, and the difference of phase is 
(2m H- l)ir. Let us now consider the case of three slits. Obvi- 
ously maxima will occur for the same values of <p as for two slits, 
i.e., where the difference of path through successive slits is mX. 
These are called principal maxima. But the minima will not 
occur midway between these as in the case of two slits. The 
reason is that, the difference of phase between corresponding 
elements of disturbance from successive slits being at the mid- 
point (2m + I)TT, two of the elements will cancel each other, and 
the third will give rise to a maximum. This series of maxima, 
midway between the principal maxima for three slits, will not 
have an intensity comparable to that of the principal maxima, 
and are called secondary maxima. On either side of these 
secondary maxima will occur minima at values of <p for which the 
disturbances from all three slits have a phase difference such 
that their sum is zero. 

These results can be described graphically by an adaptation 
of the vector polygon method described in Sec. 6 of this chapter. 
Consider each slit to be the source of a Huygens wavelet which 
has the usual characteristic of sending light in all directions, but 



SBC. 12-12] 



DIFFRACTION 



195 



with a maximum intensity in the direction of the incident wave. 
Thus each slit contributes an element of amplitude to the dis- 
turbance in a particular part of the diffraction pattern. For 
each element e\ t e z , e$, let amplitude vectors v\, v 2 , v$ be drawn, 
with the angles 'between them corresponding to a difference of 
phase which will be different at different points in the pattern. 
Then the results are as shown in Fig. 12-25. At point D of the 
diffraction pattern, for instance, at which the path difference 
between successive elements is X, and the phase difference 27r, 
the vectors are all in the same straight line, and add up to give 
the resulting amplitude of the first-order maximum. This 



Central 
image 



Secondary 
maximum 



First 
order 




Pathdiff. 
those tiff. 



Amplitude 



A 



FIG. 12-25. 



amplitude squared gives the intensity graphically represented 
above D. Similarly, at point A the path difference between 
successive elements is X/3, and the phase difference is 2ir/3, so 
that the vectors form a closed polygon. This corresponds to 
zero amplitude and intensity. At B, the path difference between 
successive elements is X/2 and the phase difference v, hence 
three such vectors give a resultant amplitude corresponding to 
the disturbance due to one element. As the angle <f> increases, 
the intensity diminishes, the graph of the entire pattern being 
enclosed by a diffraction curve exactly as in the case of two slits. 
Similar results may be obtained for more than three slits. In 
Fig. 12-26 are photographs of the patterns for 1, 2, 3, 4, 5, and 
6 equal and equidistant slits. These photographs were made 
with gratings in which the ratio of opaque to open space was 
3:1. Theory predicts that when this ratio is a whole number, 
there will be missing interference maxima which will occur at 



196 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 

points corresponding to values of <p for which the enclosing 
diffraction curve has zero height. This can be seen from eq. 12-8 
for two slits, but it will be equally true for any number of equi- 




FIG. 12-26. Diffraction through 1, 2, 3, 4, 5, and 6 slits. In each case the central 

bright portions are much overexposed. 

distant slits when the ratio of opaque to open space is a whole 
number. In eq. 12-8 when 6 * 3a, 



-f- 

COS - r - = COS 

A A 

The condition for a minimum in the enclosing diffraction pattern 



SBC. 12-13] DIFFRACTION 197 

is that va<p/\ T, but in this case cos (4^ra<f>/\) cos 47r, which 
corresponds to a maximum in the interference pattern. These 
orders are actually missing in the photographs. Also the num- 
ber of secondary maxima is N 2, where N is the number of 
openings. Their intensity relative to the principal maxima 
decreases as N increases. This is not clearly evident in the 
photographs, which were printed from the negatives so as to 
suppress the principal maxima, the latter being much overexposed 
in the negative. The most important thing illustrated in this 
series is that as N increases the principle maxima, or orders 
become increasingly sharp. For very large values of N t the 
intensities of the secondary maxima are practically zero, and 
each principal maximum (with a perfect grating) is a sharp 
image of the slit. 

These results are for monochromatic light. If more than one 
wave-length is present in the source, each order will consist of a 
spectrum. Since <p m\/(a -f- 6), it is evident these the spectra 
will have the blue end nearer the central image, i.e., at smaller 
values of <f>, than the red end. 

Diffraction gratings in practice are made by ruling lines close 
together with a diamond on polished metal or glass surfaces. In 
most cases the entire surface retains its reflecting qualities, the 
rulings serving merely to create a 
surface with a periodic structure. 
It might seem at first as if the funda- 
mental concept of diffraction does 

not hold in such a case, since the \/\/\ / \ / 
incident waves are not interrupted -\ -\' \' */ 
or cut off by obstacles in the form 
of opaque spaces. However, the 
periodicity of the reflecting surface FlQ - 12 ' 27 * 

consists of regularly spaced strips, of a width appropriate to the 
wave-lengths of the diffracted light, which are as effective in 
giving rise to regularly spaced elements of disturbance as in the 
case of a transmission grating. This is illustrated in Fig. 12-27. 
Moreover, by shaping the diamond tool with which the grating 
is ruled, it is possible to send a preponderance of light into 
particular orders. 

13. The Dispersion of a Grating. In eq. 12-9 the position 
of a maximum was given by 




198 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 

wX !/ 

-rr = sin i + sin 6. 

While this equation was derived for the case of two equal slits, it 
holds equally well for the positions of the principal maxima in the 
case of many equal and equidistant slits, since, as shown in the 
preceding paragraphs, the effect of increasing the number of 
slits is to sharpen the principal maxima, or orders. If the grating 
is made up of equidistant openings, or rulings on a reflecting 
surface a distance s apart, we may write 

sin * + sin 9 = - (12-23) 

8 

In Sec. 8-2, the dispersion D of a prism was defined as di'/d\, 
where di' is the difference of angle of dispersion of two spectral 
lines obtained with a prism. Similarly we may express the 
dispersion of a grating by 

dO 



d& being the difference of angle of diffraction between two close 
spectral lines. Differentiating with respect to X the function in 
eq. 12-23, it follows that for i constant, 



D = . _ , (12-24) 

s cos 6 ^ J 



by which it appears that the dispersion of a grating is directly 
proportional to the order of the spectrum and inversely proportional 
to the grating space. That it is independent .of the number of 
rulings has already been shown. 

^14. Resolving Power of a Grating. By definition, the resolv- 
ing power of any dispersive instrument is given by X/rfX, where dX 
is the smallest difference of wave-length which can be observed 
at the wave-length X. According to Rayleigh's criterion for 
limit of resolution, this'smallest observable d\ corresponds to the 
angle between the* maximum of a spectral line of wave-length X 
and the maximum of one of wave-length X + d\, when the latter 
coincides in position with the first minimum on either side of the 
maximum of X. But we have seen that as the number of grating 
elements (slit openings or rulings, as the case may be) is increased, 



SEC. 12-15] DIFFRACTION 199 

the maxima become narrower, and the position of the minimum on 
either side of the principal maximum (a spectral line) becomes 
closer to the center of the maximum. For instance, in the case 
of six slits the distance between one principal maximum and the 
next is the same as for two slits, but the distance between each 
principal maximum and the adjacent minimum is one-third 
as great as for two slits. For N slits, the distance between a 
principal maximum and an adjacent minimum may be seen to be 
2/N times the corresponding distance for two slits, the latter 
being X/2s when a + b in eq. 12-9 is replaced by s. Thus the 
angle of diffraction corresponding to this distance becomes, for 
N slits, 

It ' ^s = ^ sin ^ = COS 6 d6 ' (12-25) 

The resolving power may now be obtained from the relations in 
eqs. 12-24 and 12-25; i.e., 



f^ A (Itr A Tfl -m r ^ -. r /-grk r%/-\ 

R = -jr = -jr -js = - - Q Ns cos 6 = mN . (12-26) 
d\ d\ dd s cos 6 ^ ' 

Thus the resolving power of a grating is the product of the order 
of interference m and the number of rulings N on the grating, 
and is independent of the grating space. 

The resolving power may also be obtained by applying the 
general principle illustrated in Sec. 8-3, wherein it was shown that 
the resolving power of a prism may be obtained by multiplying 
the dispersion by the width a of the beam of light intercepted 
by the prism, or, by putting R = aD. Applying this principle to 
the case of the grating, for which, if w be used for the width of 
the diffracted beam and I the length of the grating, 

w I cos 6. 
Thus, 

771 

R = w - D = / cos 6 



K COS 6 

in which l/s is the number of rulings on the grating, so that 

R = mN. 

15. The Echelon. This instrument, invented by Michelson, 
is an interesting illustration of the application of the principles of 



200 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 



both diffraction and interference. Like the diffraction grating, 
it is an "interference spectrometer," belonging to class A (see 
Sec. 10-4), in which there is a division of the wave front. It 
consists of a pile of plane-parallel plates of equal thickness, 
arranged as illustrated in Fig. 12-28, each plate projecting beyond 
the one following it by a small width w. Consider a plane wave 
front F, advancing toward the right. Upon reaching any one of 

the surfaces such as S\ t an ele- 
ment of the wave front of width 
w passes through air, while 
another element of the same 
width traverses the thickness t 
of the glass plate between sur- 
faces Si and S%. The diffrac- 
tion pattern will be spread over 
a very small area if, as in prac- 
tice, w is of the order of mag- 
nitude of 1 mm. . At an angle 
of diffraction with the normal 
to the face of the glass plate the. path difference between the 
light through glass and that through air is n(cd) ab. But 




>-$.- 



T 
W 

i_ 



Sj S f 
FIG. 12-28. 



db t cos B w sin 0, 
which for small angles may be written 

ab = t wB. 
The path difference at angle 6 is thus 

(n l)t -h wO = wX, 



(12-27) 



(12-28) 



where m is the order of interference for a single plate at angle d. 
The dispersion d&/d\ obtained from eq. 12-28 is given by 



W 



JL 

w d\ 



(12-29) 



In order to express D in terms of measurable quantities, it is 
desirable to eliminate m between eqs. 12-28 and 12-29. Since 
is small, we may for this purpose write eq. 12-28 in the form 



m s 



SBC. 12-15] DIFFRACTION 201 

Substituting this value of m in eq. 12-29, we obtain 

!,.[(. -D-X*]^. (WO) 



The resolving power is given by the product of the total 
aperture and the dispersion (see Sec. 8-3). The total aperture a 
is the product of the number of plates N and the width w of each 
step, so that 



Nwt 





For small angles, (n - 1) = md/t. Substituting this value of 
(n 1) in eq. 12-31, we obtain 

R = mN - Nt~ (12-32) 

The second term on the right-hand side is small compared to the 
first, so that to a high degree of approximation, R = mN, the 
same as for the grating. 

The appearance of the spectrum will, however, be totally 
different from that ordinarily obtained with a grating. For, 
note that the smallest difference of path obtainable is that intro- 
duced by one of the plates. For small 0, m (n l)t/\, so 
that if t 1 cm., X = 5 X 10~ 6 cm., and n 1.5, then 
m = 10,000. Thus the orders observed are always very high, 
and since the angle of diffraction is small they will be very close 
together, i.e., the orders will overlap in such a way as to make 
observations on an extended region of the spectrum impossible. 
In order to avoid the overlapping of orders, auxiliary dispersion 
with a prism is used to isolate a spectrum line to be examined. 
The prismatic and echelon dispersions are in this case parallel, 
instead of at right angles, as in the case of the Fabry-Perot 
interferometer. It is customary to pass the light first through 
the slit, collimator and prism of an ordinary spectrometer, and 
then through the echelon, which is inserted between the prism 
and the camera or telescope objective. Since the intensity is 
distributed over a very narrow region, because of the small width 
of the central diffraction minimum (i.e., large w), only a few 
orders are seen at one time. For instance, with the dimensions 
given above, a 30-plate echelon would yield only three orders of 



202 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 

comparable intensity, their numerical values being of the order of 
magnitude of 300,000, say, 299,999, 300,000, and 300,001. 
Because of the extremely high order of interference possible, 
the echelon is particularly useful in the analysis of fine structure 
of spectral lines (see Chap. XIV). 

If made of glass, the echelon cannot be used to examine the 
ultraviolet below 3500 angstroms. A quartz echelon will give 
fair transmission to 1800 angstroms. In recent years, reflection 
echelons have been constructed, the steps being coated with 
reflecting metal, so that the instrument may be used for very 
short wave-lengths. For regions below 2200 angstroms it is, 
of course, necessary to place the entire instrument in an evacuated 
chamber. 

The principal drawback to the echelon is the practical diffi- 
culties involved in its construction. The plates must be of glass 
of highest optical quantity, if used for transmission, and their 
thicknesses must be as nearly the same as it is possible to make 
them. Another obvious disadvantage is that it is capable of no 
modifications in dimension to suit special conditions, as are the 
Michelson and the Fabry-Perot interferometers. 

16. Rectangular Opening. The analytical expression for the 
diffraction of a slit of width a derived in Sec. 12-8 took no account 
of the length of the slit, that dimension being considered to be 
so great that the resulting diffraction was negligible. If the 
opening, instead of being very long, has a length comparable to 
its width, the expression for the elements of disturbance after 
diffraction must contain terms taking account of both directions. 
Instead of eq. 12-1, we may write 



ds' = c sin 27r ^ - yi ~ J \dx dy, (12-33) 

L J 

where <pi = sin ii + sin 0i, and ^2 = sin *2 + sin 2 , the sub- 
scripts referring to the width a and length 6 of the opening. Then 

+a/2 / -H6/2 

S' c sin 27T~ 




c cos 2ir^ I I sin 27rl yi ~ ' yzy Ida: di/. 

f J-o/2 J-6/2 \ A / 

Since the integral of the sine between limits with the same value 
but opposite sign is zero, we can write 



SEC. 12-17] 



DIFFRACTION 



203 



Q' , in 

S = c sm 



c cos 



4 f*+<*/% (*- 

41 * I 

* J -a/2 J- 

j_ r +a/2 r 

TJ-a/2 J- 



cos 



sm 



cos 



sm 



-fc/2 X X 

The second term is equal to zero, and the first is 



dx dy. (12-35) 



S' = abc sin 2ir 



the intensity is given by 



* 



sin 



sm 



(12-36) 



sm 



x sm ' x 



(12-37) 



The resulting diffraction pattern for monochromatic light is 
shown in Fig. 12-29. 




Fro. 12-29. Diffraction by a rectangular opening whose height is five-thirds 
its width. The brightest central image and those next to it are much over- 
exposed, while of the orders, observed visually, which should complete the 
rectangular lattice, only four appear in the photograph. 



. Diffraction by a Circular Opening. While the resolving 
power for a long slit opening of width a is given by 8 = X/a, 
that for a circular opening is given by 6 = 1.22 X/a, where in 
each case 6 is the angle between the center of the diffraction 
pattern of a point object and the first minimum of intensity. 



204 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 



This can be shown in the following manner: Consider a plane 
wave advancing in a direction normal to the plane of a circular 
opening AB, of radius r (Fig. 12-30a). It is required to find the 
expression for the intensity in the diffraction pattern at an angle 
0. The resulting diffraction will be like that obtained by a lens 
which brings all rays diffracted at a common angle to a focus. 




(a) (b) 

Fio. 12-30. Diffraction through a circular opening, (a) Side view of opening 
with wave-front advancing in a direction normal to the screen. (6) Front view 
of the opening. 

The disturbance at this point will be due to the addition of all 
the elements over the area of the opening. The path difference 
between the element ds at C and that at A will be AC sin B, and 
the phase difference, 2irAC sin (0/X), so that if the disturbance at 
A is S = sin (2vt/T}, after diffraction that due to an element ds at 
C will be 



S' - 




AC sin 



jds. 



(12-38) 



But light diffracted at the angle through an element ds' (Fig. 
12-306) has the same path difference as regards the light from A 
as has that from ds, since ds' lies on a perpendicular to the line 
AB at C. Since AC r -f p cos ?. and since also the element 
ds' has an area p d<p dp in polar coordinates, the disturbance from 
ds' may be written 



SBC. 12-17] 



DIFFRACTION 



205 



ds' = sin 2?r 



ft r sin 6 p cos <f> sin 0\, , ,, rt rt/xv 

( ^ -- x -- - ^ - r p ** (12 " 39) 



and the total disturbance at the angle is 



cos 



~ cos 



in which M = - 



and AT = t 



(12-40) 
The integral 



of the second term, being an even function, is zero, so that 

p cos <p sin , , 
-- dpd<p. 



Lt/ 



. / t sin 0\ (* fr . t 
sin 27rf ~ r J I I cos 2x- 



(12-41) 

The integration with respect to <p must be carried out in series 
and that with respect to p by parts, 1 giving as the final result 
for the intensity: 



7 

7 - 



, , 

- n + - + 



VA 4 . I 2 
^J -, etc.J , 



in which n = (wr/X) sin 6. The series in the brackets, which may 
bo denoted by , is convergent for all values of w, and goes through 
positive and negative values alternately as n increases. There 
will accordingly be maximum values corresponding to ds/dn = 0, 
and zero values when s = 0. The maxima and minima in 
the resulting circular diffraction pattern, whose center will be 
on the normal to the opening, are thus at positions for which 
sin = nX/irr, where for given values of r and X, n/w takes the 
values in the following table: 





n/ir 


Intensity 


1st max 





1 


1st rain 


0.61 





2d max 


0.81 


0.0174 


2d min 


1 116 





3d max 


1.333 


0.0041 


3d min. etc . . . 


1.619 






1 For the steps in the integration see Preston, "Theory of Light," 4th ed., 
Macmillan. A discussion is also to be found in Meyer, "Diffraction 
of Light, X-Rays, and Material Particles," Appendix C, University of 
Chicago Press. 



206 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XII 

Since n is inversely proportional to X, the minima for shorter 
wave-lengths will be rings of smaller diameter. Likewise, since 
n is proportional to r, the radius of the opening, the minima will 
become smaller in diameter as the aperture is increased in size. 
Thus the size of the ring for the first minimum of intensity will be 
very small for a large telescope, and the resolving power will be 
correspondingly large. The resolving power, ao.oordincr to 




Fro. 12-31. Photograph of diffraction pattern obtained with a circular opening. 

The central image is much overexposed. 

Rayleigh's criterion, is given by sin 6 = n\/irr, in which the 
value W/TT = 0.61 is substituted, giving for small angles 

6 = 0.61-- 
r 

Problems 

1. What is the diameter of the central image, i.e., the diameter of 
the first dark ring, formed on the retina of the eye, of a distant point 
object? Assume the wave-length 5500 angstroms, and consider the 
diameter of the exit pupil of the eye to be 2.2 mm. and its distance from 
the retina to be 20 mm. 

2. Because of atmospheric disturbances, it is rarely true that the 
diffraction pattern of a star is seen distinctly; instead, the star image 
may be twice the size of the central image of the diffraction pattern 



DIFFRACTION 207 

predicted by theory. If this is true, how far from the center of the field 
3f the 100-in. telescope will the effects of coma first become visible? 
The focal length of the telescope is to be taken as 45 ft. 

3. If the headlights of a car are 6 in. across and 3 ft. apart, how close 
must an observer with normal eyes be to distinguish them as separate 
objects? 

4. In making the upper half of Fig. 12-146, a yellow filter was used 
which transmitted all wave-lengths greater than 5000 angstroms, a 
mercury arc being used. In making the lower half, only the green line, 
5461 angstroms was used. Count the number of interference fringes 
between the points of minimum visibility and calculate the mean wave- 
length of the additional radiations effective in making the upper half 
r)f the illustration. 

6. How must a grating of alternate transmitting and opaque spaces 
be constructed so that every third order will be "missing"? 

^6. A diffraction grating has 15 cm. of surface ruled with 10,000 
rulings per centimeter. What is its resolving power? What would be 
the size of a prism of glass for which B = 1.1 X 10~ 10 cm. which would 
give the same resolving power at 5500 angstroms? If the mirrors of a 
Fabry-Perot interferometer have a reflecting power of 80 per cent, what 
must be their separation to obtain the same resolving power as the 
grating? 

7. The spectrum lines formed by the concave diffraction grating are 
astigmatic images of the slit. The equations for the primary and 
secondary focal distances from the grating may be obtained from eqs. 0-8, 
for a single surface, by putting n = 1, n' = I, and changing the signs 
of s\, $2', and r. In place of the angle of refraction i' is to be used the 
angle of diffraction 6. Find the values for $/, s 2 ', and the value of 
the astigmatic difference s% /. Show that on the normal the length 
of the astigmatic spectral lines is given by / sin i tan i, where / is the 
length of the rulings 011 the grating. 

8. Describe four ways of obtaining the absolute value of a wave- 
length of light. 

9. Describe four methods for obtaining the ratio of two wave-lengths. 



CHAPTER XIII 

POLARIZATION OF LIGHT 

Thus far for none of the phenomena described has it been 
necessary to assume that the light is a wave motion of a particular 
sort. The explanations given for diffraction and interference 
will hold equally well for longitudinal waves, in which the oscilla- 
tory motion is in the direction of propagation; for transverse 
waves, in which the oscillations are at right angles to the direction 
of propagation; and for waves having a composite motion like 
that of surface waves in water. The phenomenon of polariza- 
tion, however, requires for its explanation the hypothesis that 
the vibrations are transverse. 

1. Polarization by Double Refraction. Although double 
refraction of light in crystalline media was observed by Bartho- 
linus in 1669, the first comprehensive investigation of the phe- 
nomenon was made by Huygens in 1690. He observed that on 
passing through a crystal of Iceland spar (calcite), light was doubly 
refracted, i.e., the beam was divided into two, whose separation 
depended upon the thickness and orientation of the crystal. 
From certain elementary experiments he concluded that the two 
rays had properties related to two planes at right angles to each 
other, one of them containing the crystallographic axis. Huygens 
gave to the phenomenon jbhe name polarization. 

The property of double refraction is possessed by all except 
cubic crystals. It is also a property of some organisms under 
strain. Since Iceland spar shows the property to a marked degree 
it is used extensively for experimental purposes, and offers a 
convenient medium for study. {Calcite (crystallized calcium 
carbonate) has planes of cleavage in three directions, forming a 
rhombohedron. Each obtuse angle in each plane is 101 55' 
and each acute angle 78 5'.) The form of the crystal is shown 
in Fig. 13-1. At each of the opposite corners A and A' are three 
equal obtuse angles. The line AC is an axis of symmetry with 
respect to these three faces and its direction through the crystal 
is associated with important optical properties Suppose the 

208 



SBC. 13-1) 



POLARIZATION OF LIGHT 



209 



crystal to Be placed .with its face A'B' on a screen with a pinhole 
in it to admit light ffom beneath. On looking down into the 
face AB not one image but two will be seen. Obviously these 
are due to two beams which travel in the crystal with different 
angles of refraction. More conveniently a black dot on a sheet 
of white paper may be used instead of the pinhole. The following 
observations may be made: 

a. No matter how the crystal is turned about an axis per- 
pendicular to the paper, a line drawn through the two images of 




i Side View 



FIG. 18-1. Planes of cleavage and direc- 
tion of optic axis of calcite. 



Dot 
Fi. 13-2. 



the dot will be parallel to the projection of AC (Fig. 13-2) on the 
surface of AB, as shown in Fig. 13-2a, in which the obtuse angles 
are the same in Figs. 13-1 and 13-2. Figure 13-26 shows the 
manner in which the two beams pass upward through the crystal. 

6. (As the crystal is turned about a vertical axis, the image 
toward A remains stationary. This image corresponds to the 
ordinary ray, for which the crystal acts like an isotropic medium 
such as glass or water. ( I The other image rotates about the first 
as the crystal is turned, and its position is such that the ray must 
be bent away from the normal in contradiction to the ordinary 
law of refraction. This ray is called the extraordinary ray\ 

c. The dot corresponding to the ordinary ray appears closer 
to the top of the crystal than that for the extraordinary ray. 



210 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIII 



d. If the crystal is tilted up on the corner A' (Fig. 13-1), the 
two dots draw together. If the two corners A and A' were 
flattened and polished in planes perpendicular to AC, only one 
image would be seen when viewed perpendicularly, no matter 
how the crystal was rotated about AC, as if the light were 
in an isotropic medium. The direction AC is called the optic 
axis of the crystal; it is not a particular line, but a direction 
through the crystal. 

e. If the crystal were to be flattened and polished in two planes 
parallel to the optic axis and the dot viewed perpendicularly 
through these, in general only one image would be seen as in d. 

/. If two calcite crystals are placed one above the other above 
the dot, and the top one rotated about a vertical axis as indicated 
in Fig. 13-3a, 6, and c, the images will appear as shown. 





2 B, 



(a) (b) 

FIG. 13-3. Two crystals of calcite superposed, (a) Axes coinciding; (6) axes 

at an' acute angle; (c) axes opposite. 

These observations may be explained as follows : ^Consider a 
section through the crystal which contains both the ordinary 
and extraordinary rays. It will also contain the optic axis 
and will be perpendicular to the upper and lower cleavage 
planes. This section is called a principal plane. 1 1 In Fig. 13-4 
is shown a principal plane as the plane of incidence, the line AC 
being the same direction as in Fig. 13-1 and 13-2. A parallel 
beam of ordinary light passing up through the crystal from a dot 
on the lower cleavage plane is divided into two beams which 
travel through the crystal in different directions and with 
different velocities. 

J For the ordinary ray traversing the crystal in any direction, 
the Huygens wavelets will be spherical in shaped In Fig. 13-4 

\] In optical mineralogy a principal section or plane is one containing the 
ray and the optic axis of the mineral.^ 



SBC. 13-2] 



POLARIZATION OF LIGHT 



211 



these wavelets are represented by small circles, the common 
tangent of which will be the wave front. I The perpendicular to 
the wave front is the direction of the ordinary ray. ( The extraor- 
dinary ray travels through the crystal with a velocity which is the 
same as that of the ordinary ray 
in the direction of the optic axis, 
and which becomes increasingly 
greater as its "direction of propa- 
gation makes~larger angles with 
the axis, until its maximum 
velocity occurs perpendicular to 
the axis. This is shown by the 
variation In the appearance of 
depth of the two refracted im- 
ages which is described in experi- 
ment (c) above. In order to 




FIG. 



13-4. The passage of the ordi- 
nary ray through calcite. 



represent the propagation of the 
extraordinary ray, the Huygens wavelets must be drawn as 
ellipses, l as in Fig. 13-5, with the long axis perpendicular to the 
optic axis of the crystal. The wave front of the extraordinary 
ray will be the common tangent of the ellipses. While the wave 




FIG. 13-5. The passage of the extraordinary ray through calcite. 

front remains parallel to itself, the ray is not normal to it, thus 
acting in a manner contradictory to the ordinary laws of refrac- 
tion. If the light should be incident upon the crystal at such 
an angle that its direction through the crystal is parallel 
optic axis, there will be only one ray. 

2. The Wave-velocity Surface. In the foregoing it is 
posed (1) that the light is incident normally to the surface of the 



212 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIII 

crystal, and (2) that the optic axis is parallel to the plane of 
incidence. If (2) holds but (1) does not, the resulting refraction 
is as illustrated in Fig. 13-6. Here the plane of the paper is the 
plane of incidence. The optic axis is parallel to this plane, and 
M N indicates the intersection with the surface of the crystal 
which is perpendicular to that plane. To find the path of a 
plane wave, MM', incident obliquely on the face of the crystal, 
we may proceed as follows: By the usual Huygens construction, 
a circle is drawn with the center at M and a radius equal to 
M'N/riQ, where n is the index of refraction of the ordinary ray. 
The line drawn from N tangent to this circle will be the refracted 




FIG. 13-6. Optic axis not in refracting surface of crystal. 

wave front for the ordinary ray, and MO drawn through the 
point of tangency is the direction of the ray. Since the velocity 
of the extraordinary ray is greater than that of the ordinary in 
directions other than that of the axis, the Huygens construction 
will be an ellipse touching the circle at the axis and having^ a 
semimajor axis equal to M'N /n e . The tangent from N to this 
ellipse is the extraordinary wave front, and Me drawn to the 
point of tangency is the ray. 

t. The laws of refraction in ordinary isotropic media were first 
stated by Descartes as follows: "The incident and refracted 
rays (a) are in the same plane with the normal to the surface, 
(6) they lie on opposite sides of it, and (c) the sines of their inclina- 
tions to it bear a constant ratio to one another." It is evident 
that for the case just described (c) is not obeyed, for the ratio of 
the sines of the angles for the extraordinary ray will vary with the 
angle of incidence./ If the optic axis is not in the plane of inci- 



SBC. 13-2J 



POLARIZATION OF LIGHT 



213 



dence but making an angle with it, in general the point of 
tangency of the extraordinary wave front to the ellipse will not 
be in the plane of incidence and (a) will also be violated. 





Fiu. 18-7. Optic axis perpendicular to the page. 

A special case is illustrated in Fig. 13-7. Here the optic axis 
is perpendicular to the plane of incidence. Since the velocity 
of the extraordinary ray is a maximum in every direction per- 





FIG. 13-8.- -Optic axis parallel to the refracting surface and to the page. 



pendicular to the axis, the Huygens construction for each ray is a 
circle. In this case all the ordinary laws are satisfied. 

If the optic axis is parallel to the face of the crystal and also 
parallel to the plane of incidence, as shown in Fig. 13-8, an inter- 
esting relation exists. A line dropped from T e , the point of 
tangency of the extraordinary wave front, perpendicular to MN, 



214 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIII 



will pass through To, since the polar 1 of any point such as N on 
the chord of contact of a circle and an ellipse is the same for both 
curves. When an ellipse is projected into a circle, the ratio of two 
lines such as T e D and T D is the same as the ratio of the semi- 
major to the semiminor axis; hence it follows that 



tanr = T f D 
tan r e T Q D 



n 



(13-1) 



where n e is understood to correspond to the maximum velocity 
of the extraordinary ray through the crystal. This relation and 




FIG. 13-9. 

others which may be found by similar constructions have been 
experimentally verified, supporting the assumption that the 
surface of the Huygens wavelet for the extraordinary ray is an 
ellipsoid of revolution formed by revolving an ellipse about its 
minor axis, which is parallel to the optic axis. 
I Of great assistance to an understanding of the manner in 
which the two rays traverse the crystal is a model of the wave- 
velocity surface. In the case of calcite this will consist of a sphere 
inside an ellipsoid and tangent to it at the extremities of the 
minor axis./ A very satisfactory model, illustrated in Fig. 13-9, 



SEC. 13-4| POLARIZATION OF LIGHT 215 

may be made of three pieces of cardboard, one circular, the other 
two elliptical, fitted together at right angles. On these may be 
indicated by circles or colored areas the wave-velocity surface 
of the ordinary ray. The model represents, of course, the dis- 
tances traversed in a given time by the light from a point source 
inside the crystalline medium. 

3. Positive and Negative Crystals. Uniaxial Crystals. 
Caicite is one of a group of crystals which possess a single direc- 
tion in which the wave-velocity surfaces are tangent to one 
another.l These are called uniaxial crystals. 1 * \ Crystals in which 
the common tangent to the two wave fronts corresponds to more 
than one direction through the crystal are called biaxial, n Uni- 
axial crystals may be further divided into two groups, depending 
on whether the velocity of the extraordinary ray is greater or 
smaller than that of the ordinary. Caicite belongs to the former 
class and is called a negative crystal, while those of the latter are 
called positive. \ \ The most useful positive uniaxial crystal is 
quartz,| since it occurs in abundance in many places, is hard, and 
transmits a considerable portion of the near ultraviolet in the 
spectrum. I The wave model for a positive crystal will consist 
of a sphere outside an ellipsoid of revolution about the major p.xis 
of the ellipse, which would be equal to the diameter of the sphere A 

The indices for a partial list of positive and negative crystals 
are given in Table II at the end of the book. 
\ In addition to the property of double refraction, some crystals 
also absorb the two rays unequally. In tourmaline, an aluminous 
silicate of boron containing sodium, magnesium, or iron, one of 
them is absorbed completely, so that if two plj^es are ^ut from 
it with their faces parallel to the optic a*^ .,. ^pn 

crossed, extinguish the light completely. Recently some success 
has been achieved in the preparation of crystals of quinine-iodine 
sulphate in thin sheets of transparent material. These have 
properties similar to those of tourmaline, so that pieces cut from 
the same sheet and crovssed may be used with considerable success 
in experiments in polarized light. This material, called polaroid, 
is also useful for the reduction of glare due to light reflected from 
polished surfaces, since such light is often polarized^ 

4. Polarization by Reflection. In 1808 Malus discovered that 
after reflection fron^ the surface of a transparent substance such 
as glass the light exhibited the same properties of polarization 



216 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIII 



as* the separate beams transmitted through doubly refracting 
crystals. This can be demonstrated in the following way: From 
a source (Fig. 13-10a) allow a beam of light to pass through a 
horizontal slot about 5 mm. high to a clean glass plate Pi at an 
angle of incidence of about 57 deg. Above Pi place another glass 
plate P 2 which can be rotated about a vertical axis. It is con- 
venient to exclude light from other sources by laying beneath Pi 
a piece of black cloth or paper. If possible. P 2 should be made of 
black glass, but if this is not available, a backing of black paper 
may be gummed to it, or it may be coated on the back with 
optical blacky On looking into P 2 from the position indicated in 

f 

i 






- Pile of 
\ plates 




FKJ. 13-10. 

Fig. 13-10a, no great change in intensity will be observed in the 
light from S as P 2 is rotated about a horizontal axis, but if it is 
turned through 90 deg. about a vertical axis, it will be found that 
afterward a rotation about a horizontal axis will cause the light 
to. chan*^ ; " intensity. When the angle of P 2 is such that the 
pj' .V AAsiice upon it is at right angles to the plane of inci- 
dence upon Pj, and the angle of incidence of the ray upon P 2 is 
about 57 deg., the light from S will be extinguished. The angle 
of incidence upon either mirror is then called the polarizing angle. 
This experiment can be explained by certain assumptions as to 
the nature of the light and the effect upon it of reflection at the 
two mirrors. Let us suppose that the light is a transverse wave 
motion, composed of vibrations in all possible orientations in 
planes perpendicular to the direction of propagation. Then 
these vibrations may be resolved into two sets of components 
perpendicular to each other, as shown in Fig. 13-11, in which the 
short bars crossing the rays represent components of vibration 



SEC. 13-4] 



POLARIZATION OF LIGHT 



217 



in the plane of incidence and the dots, components perpendicular 
to that plane. Let us assume also that, upon reaching the glass, 
the light passes into it, has its direction changed by refraction, 
is partly absorbed and a part of the absorbed light is re-emitted 
in the reflected ray. If the angle between the refracted and 
reflected rays is 90 deg., then no part of the components of vibra- 
tion parallel to the plane of incidence in the former can be 
re-emitted in the latter, since light is assumed to be a transverse 
vibration. At the mirror P* in the position of Fig. 13-106 none 
of the plane-polarized light will be 
reflected, provided the angle is 
the polarizing angle, since its direc- 
tion of vibration is parallel to the 
plane of incidence in Pa. 

The ray refracted into the glass 
P\ t however, will consist of light 
resolvable into both components 
of vibration just as was the origi- 
nal beam, but with a reduction 
of the amplitude of the component 




perpendicular to the plane of inci- , J 10 ' i3-ii. -^hematic 

* r i tation of polarization by reflection. 

donee. This suggests another The light in the incident beam is not 

experiment. For, if instead of a actual | y broken up ; n , to * w c P m ' 

K ' ponents as represented, but, since 

single plate of glaS a pile of plates upon reflection the amplitudes paral- 

thpn at pap lel and Perpendicular to the plane 

L11CI1. rtl/ VjCtCl 



, . . , j i i- i 

_ of incidence are to be discussed 

refraction there will be a reduction separately, these components are 

of amplitude of the component per- also &t** in incident beam. 



pendicular to the plane of incidence, provided the pU/pJ^ a 
polarizing angle or nearly so. Twelvo or nfteen plates \*rith 
clean and sensibly plane surfaces will extinguish that component 
of the refracted light completely. . In Fig. 13-10c is shown a 
pile of plates which is set for extinction. 

In a sense, however, the term "extinction" is a misnomer here, 
for the intensity of the light is ordinarily only reduced to a 
minimum at the polarizing angle. Jamin found that only for 
certain glasses whose index is about 1.46 is the polarization ever 
complete. In general, it may be said that the polarizing angle 
is the angle of most complete polarization. 

These experiments support the theory that light is a transverse 
vibration. No analogous results can be. obtained with longi- 



218 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIII 



tudinal sound waves. The hypothesis which has been introduced 
concerning the nature of the mechanism of reflection has been 
found by other experiments to be sound and is supported by 
accepted theories of the nature of light. 

6. Brewster's Law. It is evident that if the mirror Pi (Fig. 
13-106) is set at angles other than the angle of polarization, the 
reflected beam will contain a considerable proportion of light 
whose direction of vibration contains a component parallel 
to the plane of incidence. At the polarizing angle, this is reduced 
to a minimum, and the reflected and refracted rays are per- 
pendicular to each other. Then Snell's law becomes 



n = 



sin 



sin i 



sin r sin (90 i) 



= tan i. 



(13-2) 



This result is known as Brewster's law, after its discoverer. 
The polarizing angle thus depends upon the transparent sub- 
stance used, and to a small extent upon the wave-length of the 
light. The polarizing angles for the orange and blue for two 
representative kinds of glass are given below: 



X 


Color 


Ordinary crown 


Heavy flint 


n 


i 


n 


i 


5893 
4358 


Orange 
Blue 


1.518 
1 . 529 


56 38' 
56 49' 


1.717 
1.749 


59 47' 
60 15' 



For ^.toueu metallic surfaces Brewster's law does not hold, 
although some degree of polarization occurs upon reflection. 

6. Direction of Vibration in Crystals. We are now, in a posi- 
tion to determine the direction of vibration of the two rays in 
calcite. Let P\ (Fig. 13-10a) be placed at the polarizing angle 
and in place of Pz be placed a section of calcite split along cleavage 
planes, with a dot of ink on its lower surface. It will be observed 
that when the calcite is held with the cleavage faces horizontal 
and is turned about a vertical axis, there is a position where the 
ordinary ray disappears. This will be when the principal section 
of the crystal, containing the optic axis, is perpendicular to the 
plane of incidence. If the crystal is turned through an angle of 
90 deg. the extraordinary ray disappears. Because of some 



SBC. 13-7] POLARIZATION OF LIGHT 219 

inaccuracy in adjustment of the apparatus the disappearance in 
either case may not be complete, but the reduction to almost zero 
intensity will be evident. From this experiment it is clear that 
the ordinary ray corresponds to a vibration in a plane perpendicu- 
lar to the principal section, while the extraordinary ray corresponds 
to a vibration in the principal section. At an angle of the crystal 
about halfway between the two positions the dots will be approxi- 
mately of equal intensity. This experiment throws additional 
light upon the experiment (e) in the first section in this chapter, 
where the number and intensity of the images changed as one 
crystal was rotated above another. 

7. Plane of Polarization. From his observations on double 
refraction in calcite Huygens concluded that the ordinary and 
extraordinary rays must in some way be related to the principal 
plane and the optic axis. He differentiated between the two 
rays by postulating that the ordinary ray was polarized in the 
principal plane and the extraordinary perpendicular to it. The 
existence of these so-called planes of polarization is substantiated 
by further consideration of polarization by reflection. In 
Sec. 13-4 it was pointed out that only when the two mirrors P t 
and PZ (Fig. 13-10a) are arranged so that the planes of incidence 
and reflection at both mirrors coincide is the light which reaches 
the eye a maximum in intensity. This may be interpreted in 
the form of a conventional statement that the plane of incidence 
thus described is the plane of polarization of the reflected light. 
Actually, however, from the results of experiments, some of 
which have been outlined in the preceding paragraphs, it appears 
that the direction of vibration in every case is perpendicular to 
this plane of polarization. The original phraseology of Huygens 
still persists in treatises on the subject, and plane of polarization 
is still referred to, rather than plane of vibration] in fact, fre- 
quently both terms are used. There seems to be no reason for 
using both terms in an elementary discussion of the phenomena 
of polarization of light. This text will avoid further description 
of the phenomena in terms of the plane of polarization and will 
continue to refer to the direction of vibration of polarized beams. 
It is not denied that something of importance may be taking 
place in directions other than the direction of vibration ; whatever 
it may be, however, does not come within the scope of the present 
discussion. 



220 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIII 

8. The Cosine-square Law of Maltis. When Pi (Fig. 13-10a) 
is set at the polarizing angle, a ray of ordinary light incident 
upon it and the polarized ray reflected from it define the plane of 
reflection at the mirror. If P 2 is also set at the polarizing angle 
so that the plane of reflection from it is at right angles to that from 
Pi, extinction will take place. For positions of the two mirrors 
in which their respective planes of reflection are not at right 
angles, the light will be partly reflected. (These facts were 
summed up by Malus in the statement that with the two mirrors 
set at the polarizing angle the intensity of the twice reflected 
beam varies as the square of the cosine of the angle between 
the planes of reflection^ For instance, if Pi and P 2 have the 
positions shown in Fig. 13-10a, the intensity is 7, but if P 2 is 
oriented to a position making an angle intermediate between 
those shown in Fig. 13-10a and 6, say, 60 deg. from that in a, 
then the intensity of the light reflected from P 2 is 

/' = / cos 2 60 = 7/4. 

At the position shown in Fig. 13-106, I' = 0. 

9. The Nicol Prism. As a device for producing or examining 
plane-polarized light, the glass plate used at the polarizing angle is 
lacking in convenience. Also, to obtain plane polarization over 
any considerable area it is necessary to collimate the light. 
It is usual to employ some prism of double-refracting crystal 
arranged so that light vibrations in only one plane are trans- 
mitted. One of the most convenient prisms is the nicol, named 
after William Nicol, who first described its construction in 1828. 
The original form was made of a rhombohedron of calcite about 
three times as long as it was wide. As shown in Fig. 13-12a, 
this is cut along a plane perpendicular to the shorter diagonal 
of the end face, which is diamond-shaped. The two pieces are 
then cemented together again with Canada balsam, which has an 
index of refraction intermediate between n and n e for calcite. 
Since the natural angle of the end faces is slightly altered, light 
incident upon the nicol parallel to the long sides will be refracted 
so that the extraordinary ray is incident upon the interface at an 
angle less than the critical angle of refraction, and will thus be 
transmitted with no appreciable loss of intensity, while the 
ordinary ray is incident at an angle greater than the critical 
angle of refraction and so is totally reflected. The direction of 



SEC. 13-10] 



POLARIZATION OF LIGHT 



221 



vibration of the transmitted extraordinary ray is in the plane 
containing the ray and the short diagonal of the end face, as 
illustrated in Fig. 13-126. Nicol prisms are often made in other 
shapes, to admit beams of wider angle or greater cross-sectional 
area, but the effect is the same, i.e., to transmit light with vibra- 
tions in only one plane. It is usual in examinations of polarized 
light to make use of two nicols. The one nearer the source, 




(a) 




A 





.1 J 


y ^**O^ 


i 7 


/ ^\ 

f ^^ 

/ "*V. 


\/ 

& 



B 



(b) 




FIG. 13-12. The Nicol prism, (a) Principal section; (6) side view; (c) end view, 
the arrow showing the direction of vibration of the transmitted light. 

called the polarizer, is for the purpose of producing the plane- 
polarized beam; the other, nearer the eye, called the analyzer, is 
for the purpose of examining the state of polarization of the 
transmitted light. 

10. Double Image Prisms. The Wollaston Prism. While 
the nicol transmits light vibrations in a, single plane and eliminates 
the vibrations perpendicular to that! plane by total reflection, 
it is sometimes necessary to retain both components so that the 
two separate images, polarized perpendicularly to each other, 
are in the field of view. This can be done for objects with 



222 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIII 



limited area by the use of an ordinary crystal of calcite, but with 
the disadvantage that the emerging beams are parallel and 
cannot be easily separated to any greater extent. There are 
several polarizing devices which will give two diverging beams, 
tjie v most useful probably being the Wollaston prism. Its con- 
struction and use are illustrated in Fig. 13-13. The light is 
incident normally on a compound prism of quartz with parallel 
faces made of two prisms cemented together, whose optic axes 
are perpendicular to the direction of propagation but also per- 
pendicular to each other. The ordinary and extraordinary 
beams will thus traverse the first prism in the same direction 
but with different velocities. Since the second prism is cut with 
its optic axis perpendicular to that of the first, the ordinary 
beam in the first prism will become the extraordinary in the 
second, and vice versa. At the interface between the two prisms, 

t the beam traveling faster in the 

first than in the second has an 
angle of refraction which is smaller 
than the angle of incidence. Like- 
wise, the beam which travels 
slower in the first prism than in 
the second has an angle of refrac- 
tion which is larger than its angle 
of incidence. The interface is cut 




FIG. 13-13.- The Wollaston prism, at such an angle that the two plane- 

polarized beams traversing the 

second prism are equally inclined to the emergent face. Thus 
each beam will undergo the same amount of bending by refrac- 
tion, and the two will emerge into the air at the same angle to 
the normal but oppositely inclined. The larger the distance from 
the prism, the farther apart will be the two images. 

If to a Wollaston prism is added a nicol used as an analyzer, 
the combination is known as a Cornu polariscope. The nicol 
may be rotated to an angle such that the two plane-polarized 
images are transmitted with the same intensity, in which case the 
ratio of the intensities incident upon the nicol is proportional to 
the square of the tangent of the angle. The Cornu polariscope 
is a useful device for the detection of small amounts of polariza- 
tion, since a small change of angle of the nicol results in a large 
change of the relative intensities of the two images. 



SEC. 13-11] 



POLARIZATION OF LIGHT 



223 



( 11. Elliptically Polarized Light. Wave Plates. Suppose a 
beam of plane-polarized light is incident, as in Fig. 13-14a, 
upon a thin section of crystal whose faces are parallel to each 
other. For convenience suppose also that the optic axis is 
parallel to the faces but makes an angle a with the plane of 
vibration of the incident beam. Then the original vibration 
will be divided in the crystal into two components as illustrated in 
Fig. 13-146. The component of vibration parallel to the axis 
(extraordinary ray) will have an amplitude A cos a, and that 
perpendicular (ordinary ray), an amplitude A sin a, where A 
is the amplitude of the incident vibration. If the plate is thin 
and the source of appreciable area, there will be no detectable 



M-f 




rr '/ 

I/ 









* 

' e 



7 







/ 
\_ 



(a) (b) (c) 

FIG. 13-14. The heavy arrow in (ft) indicates the original plane of vibration of 
a plane-polarized beam, and in (c) the plane of vibration after passing through 
a half-wave plate. 

separation between the two beams, but since their velocities 
are not the same, they will emerge from the crystal with a differ- 
ence in phase. If the retardation of phase (of the ordinary in a 
negative crystal, the extraordinary in a positive) corresponds to 
an even number of half wave-lengths difference of path, the plane 
of vibration of the emergent beam will be the same as that of the 
incident beam. If the retardation corresponds to an odd number 
of half wave-lengths, the two components will after emergence 
have the relative positions shown in Fig. 13-14c, and their com- 
bined effect will be that of a plane vibration in a plane making an 
angle 2a with that of the incident beam. A plate which thus 
effects a turn of the plane of vibration of the light is called a 
half -wave plate. If the angle a is 45 deg., the emergent vibration 
will be in a plane at right angles to the incident plane-polarized 
beam. 

If the retardation corresponds to an odd number of quarter 
wave-lengths, the emergent components will combine to form, 



224 



LIGHT: PRINCIPLES AND EXPERIMENTS (CHAP. XIII 



not a plane vibration, but one which in general is elliptical in foim. 
An analogue is the motion of a particle which executes an ellip- 
tical motion in a plane which is moved normal to its surface. 
If in addition the angle a is 45 deg., circularly polarized light 
results. A plate of crystal which produces these results is called 
a quarter-wave plate. 

The effect upon the passage of plane-polarized light through 
thin crystals can be best treated analytically. Let OP (Fig. 
13-15) represent the amplitude and direction of vibration of a 
plane-polarized beam traveling perpendicular to the page, and 
OX the direction of the optic axis of a double-refracting crystal, 

of which the plane cutting OX 
perpendicular to the page is a 
principal section. In the crystal 
the incident vibration will be sepa- 
rated into two, one parallel, the 
other perpendicular, to the prin- 
cipal section. After passage 
FIO. 13-15. through a crystal whose thick- 

ness is such that a path difference of 6 is introduced between the 
ordinary and extraordinary rays, the amplitude of the X- and 
F-components is given by 




n t 
x = a cos 2ir-fp> 

/ 8\ 
y = o cos 2ir[ 7= + - ) 

V V 



(13-3) 



If these two equations are combined so as to eliminate t, the 
result will be 1 

z 2 2/^ _ 2xy cos (2 

tf "*" P ^~ 



= sin 2 =p, (13-4) 

A 



which is the equation of an ellipse, representing in general the 
character of the vibration aft6r emergence from the crystal. 
This ellipse may be inscribed in a rectangle whose sides are 2a 
and 26, the ratio of the sides depending on the angle a between 
the ^original plane of vibration and the principal section OA. 

1 This may be done by solving the first equation for cos 2irt/T, expanding 
the second and solving it for sin 2vt/T; squaring and adding, making 
suitable substitutions. 



SBC. 13-11) POLARIZATION OF LIGHT 225 

The particular character of the transmitted light will depend 
upon the values of 6 and a. 

Case 1. 8 wX, n = 0, 1, 2, 3, etc., eq. 13-4 becomes 

*- - \ - 0. 
a o 

The emergent light is plane-polarized, the vibrations being in the 
same direction as in the original beam. 

Case 2. 5 = (2n + l)X/2, n = 0, 1, 2, 3, etc., eq. 13-4 

X 11 

becomes h j- 0. The emergent light is plane-polarized 

G& (/ 

in a direction making an angle 2a with the original beam. The 
original beam is in the first and third quadrants, the emergent is 
in the second and fourth quadrants. If a 45, the vibrations 




5X 





(b) 
Flo. 13-16. In (a) the angle a is w/4; in (/>) it is leHs than w/4. 

in the emergent beam will be in a plane perpendicular to those in 
the incident beam. 

Case 3. 8 = (2n + l)A/4, n = 0, 1, 2, 3, etc., eq. 13-4 

x^ y^ 
becomes -5 + j-$ = 1. The emergent beam is elliptically 

CL c/ 

polarized with the axes of the ellipse parallel and perpendicular, 
respectively, to the principal section of the crystal. If a = 45, 
a = b and 2 -f- 2/ 2 = a 2 , and the emergent light is circularly 
polarized. For 5 = A/4 the vibration corresponds to the motion 
of a particle moving in a clockwise direction in a circle of radius 
a; for 5 = 3A/4 the circular vibration will be in the opposite sense, 
i.e., counterclockwise. 

Case 4. If 5 is other than an integral multiple of X/4, and 
therefore does not come under one of the three cases above, the 
light will in general be elliptically polarized. 

For a = 7T/4, these results may be represented graphically as 
in Fig. 13-16a. The straight line on the left represents the plane 
of the emergent vibration for a plat^bf thickness zero, or the 



226 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIII 

plane of vibration of the incident beam. The others represent 
the effect on the state of polarization of the emergent light by 
passage through such crystal thicknesses as to introduce addi- 
tional path differences of A/8. The arrows indicate the direction 
of rotation in the cases of circular and elliptical polarization. 
If the angle a is less than 45 deg., the resulting vibrations will 
be as in Fig. 13-166. If a is greater than 45 deg., the effect will 
be to increase the vertical axis instead of diminishing it. 

It will be noticed that except for a = or ?r/2 it is possible to 
produce elliptically polarized light with wave plates which 
introduce a retardation of an odd number of quarter wave- 
lengths, the difference between this case and case 4 above being 
that with the quarter-wave plate the axes of the ellipse are parallel 
and perpendicular to the principal section of the crystal. It 
should also be noticed that it is impossible to obtain circularly 
polarized light with a quarter-wave plate unless the angle 
between the optic axis and the plane of vibration of the incident 
light is 45 deg.) ' 

In practice it is customary to make quarter-wave plates and 
half-wave plates of mica. Although mica is not uniaxial but 
biaxial, in some kinds the angle between the two axes is small. 
Mica vsplits easily along planes of cleavage which are perpendic- 
ular to the bisector of the angle made by the optic axes. Because 
of the strains introduced when it is made into thin sheets, cello- 
phane is also anisotropic, and may be used for making quarter- 
wave plates by superposing two pieces at a suitable angle. 

In the preparation of wave plates it is customary to mark with 
an arrow the plane of vibration of the slower component through 
the crystal; if the plate is made of calcite or some other uniaxial 
negative crystal, this is also perpendicular to the direction of the 
optic axis. In mica, it indicates only the plane of vibration of the 
slower component, or the principal section in which that com- 
ponent vibrates. 

( 12. The Babinet Compensator. In the production or analysis 
of elliptically polarized light the quarter-wave plate is limited to 
a narrow band of wave-lengths. There are several devices which 
do not have this limitation, the compensator of Babinet being 
the most useful. As illustrated in Fig. 13-17, it is made of two 
wedges of quartz with their optic axes perpendicular to each 
other, and both perpendicular to the direction of propagation of 



SBC. 13-12] 



POLARIZATION OF LIGHT 



227 



the transmitted light. As improved by Jamin, one of the wedges 
is arranged to slide with respect to the other, the amount of 
motion being controlled by a micrometer screw. In the figure 
the angles of the wedges are much exaggerated. It is apparent 
that for the ray traversing equal thicknesses in the two wedges 
there is no difference in phase introduced since each plate pro- 
duces an identical amount of retardation of the slower com- 




FIG. 13-17. Diagram of Babinet-Jamin compensator. 

ponent of plane-polarized light. Where the thicknesses traversed 
are not equal, there will be a phase difference introduced between 
the two components. The compensator is therefore at every 
point equivalent to a wave plate, and introduces a relative 
retardation between the components vibrating in planes parallel 
to the optic axes of the two wedges. Zero at the point where 
equal thicknesses are traversed in the two wedges, this retarda- 



Polciriier 



Compensator Analyzer 





Fio. 13-18. 

tion increases uniformly on either side, and is of opposite sign on 
the two sides of the zero point. 

Let us suppose that the light incident on the compensator is 
monochromatic and plane-polarized, with its plane of vibration 
neither parallel nor perpendicular to the plane of incidence. 
This may be effected by means of the polarizer (Fig. 13-18). 
In case it is desired to change the state of polarization of the 
light incident on the compensator, a quarter-wave plate may be 
inserted, as shown. In the first wedge, A, the light is resolved 
into an ordinary 'and an extraordinary beam. If the thickness 
traversed in wedge A is t\ t the relative retardation of the two 



228 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIII 



beams is t\(n e no), while for thickness k traversed in wedge B 
the relative retardation is h(n e n Q ). The total retardation 
will be 

e - n ). (13-5) 



For the light traversing equal thicknesses of A and B, 8 = 0. 
At distances from this middle position having retardations 
X, 2X, 3X, etc., the light is plane-polarized and vibrating parallel 
to the original plane, as at the middle position. At positions 
midway between these, where the retardation is X/2, 3X/2, 5X/2, 
etc., the transmitted light is plane-polarized but vibrating in a 
plane making an angle 20 with the original plane of vibration, 

Center 



(b) Path cliff. -; 


. JA A A . 

i "7 ~J < 




(c) Phase cliff. -2 


'// -/7 ( 


> +/T 


Positions 
(ef) of dark \ 
fringes * 


i 1 

N* /* -J 


1 






i 



Fio. 13-19. Polarizations due to a Babinet compensator with its axes at 
45 deg. to the direction of vibration of an incident plane-polarized beam. The 
fringe pattern (d) really has an intensity distribution similar to that of double- 
beam interference fringes and only the positions of the minima are indicated. 

where B is the angle between that plane and the direction of the 
optic axis in the first wedge. Thus there is a set of equidistant 
positions at which the light is plane-polarized, and in any adjacent 
pair of such positions the planes of vibration are parallel to, 
and at an. angle of 26 to, the original plane of vibration. At all 
other positions the light will in general be elliptically polarized. 

If B = 45, the alternate plane-polarized beams transmitted 
by the compensator will consist of vibrations at right angles to 
each other. Midway between these the retardation will be like 
that of a quarter-wave plate with its principal section at 45 deg. 
to the plane of vibration of the incident light, and the'transmitted 
light will be circularly polarized. At other positions, the retarda- 
tion will be that of a fractional-wave plate producing elliptically 
polarized light. This situation is illustrated in Fig. 13-19a, in 
which the arrows on the circles and ellipses represent the direction 
of rotation introduced. If the field of the compensator now be 



SEC. 13-12] 



POLARIZATION OF LIGHT 



229 



viewed through a nicol (analyzer) set so as to extinguish the light 
transmitted at the middle position, there will be seen a set of dark 
fringes crossing the field perpendicular to the long edge of the 
compensator. These are indicated in Fig. 13-19d Only the 
central fringe will be black if white light is used; the others will 
be colored. If 9 7* 45, the fringes will be dark, but not black, 
with monochromatic light. The distance apart of the fringes 
will correspond to phase differences of 2ir, or path differences of X. 
If the analyzer is rotated- through 45 deg., these fringes will 
disappear and the entire field will become uniform in intensity, 
since the analyzer will have no effect at the positions of circular 




Flo. 13-20. 

polarization, and at all other positions will effect a retardation 
changing plane or elliptical vibrations to circular ones. 

It is thus evident that the compensator will in general transmit 
elliptically polarized light for which the ellipticity will depend 
upon the 8 in eq. 13-5 and which will, for 6 = 45 and also 
for certain values of 5 be plane or circularly polarized. It is 
possible to obtain an equation representing the form of the 
emergent vibration for any value of 8. Let the plane and ampli- 
tude at any instant of a plane-polarized beam incident on the 
compensator be represented by OP, (Fig. 13-20) and the X- and 
F-directions represent the directions of the optic axes of the 
wedges A and B, respectively. The components of OP in the 
X- and F-directions are, respectively, 

a OP cos 0, 
6 = OP sin 0, 



230 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIII 



the values of a and 6 used here being the maximum amplitudes 
of the vibration components in the X- and F-directions. The 
vibrations themselves may be represented by 



x = a cos 2ir-~ 



y = b cos 2ir 




(13-6) 



in which 2irA is the difference of phase angle between the com- 
ponents after transmission through the compensator. It has 
already been shown by the detailed discussion that for = 45 
the axes of the elliptical vibration will be parallel and per- 
pendicular to OP, so that, in order to find the analytical form of 
the vibration, we must obtain equations analogous to eqs. 13-6 for 
the vibrations in these directions. From Fig. 13-20 it is seen 
that a may be resolved into two components, u\ in the direction 
of OP and Vi perpendicular to OP] likewise b may be resolved into 
u-i and v z . As u\ and HZ are not in general in the same phase, 
they cannot be added algebraically, nor can v\ and v z be so added; 
the additions can only be made with the proper phase relations 
assigned. This may be done by writing for the components of 
vibration in the U- and Indirections the general expressions 



U U\ COS lT 



v = v\ cos ?r~ 




taking into account a phase difference 2?rA, and substituting in 
them the values of u\ y u^, Vi, and v for the case under discussion. 
From above 

u\ = a cos 6 = OP cos 2 6, 

HI = b sin B = OP sin 2 0, 

Vi = a sin = OP sin cos 0, 

t>2 = b cos = OP sin cos 0, 



so that 

u - OP 



v = OP 



cos 2 cos 2ir-~. + sin 2 cos 2r( -^ 




sin cos cos 2ir 



( ^ H- A J si 



sin cos cos 



i] 



SEC. 13-12] 



POLARIZATION OF LIGHT 



231 



The phase number A is 6/X, or the number of wave-lengths 
difference of path in the distance 6 given in eq. 13-5, so we may 
write 



2irA = 






n ). 



(13-7) 



For the special case, = 45, the equations for u and v reduce 



to 



u 



v = 



OP 



2 
OP 




cos 



which by a simple trigonometric transformation become 



u 



cos TrA 

V 

cos 



= OPcos27r(4 + V 



= -OP sin 27T 



(I + f > 



On squaring and adding, these reduce to 



cos 2 (A/2) ' sin 2 (A/2) 



= OP 5 



(13-8) 



This gives the form of the vibrations for different values of A, 
i.e., for different positions in the field of view. If the light trans- 
mitted at the position A = is extinguished by a nicol, the inten- 
sity at all other positions is given by v* = OP 2 sin 2 (A/2),-^ 

If the light incident upon the compensator is changed from 
>lane to elliptically polarized light by means of a quarter-wave 
)late inserted between the polarizer and the compensator, in 
general there will be a shift of the fringes by an amount depending 
jn the ratio of the major and minor axes. There will also be a 
change in their blackness since nowhere will the light be plane- 
polarized in the plane extinguished by the analyzer. A rotation 
M the analyzer to the angle of extinction will restore the blackness 
rf the fringes. \ 

The value ol 2irA given in eq. 13-7 may be found experimentally. 
The entire fringe system may be moved to bring successive dark 
fringes under the cross hairs by moving wedge B. If the actual 



232 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIII 

distance between the fringes is s, the wedge must be moved a 
distance 2s to move the fringe system through the distance 8. 

If the change from plane to elliptically polarized incident light 
shifts the fringe system a distance x, the corresponding difference 
of phase introduced is TTX/S. This is, however, the amount by 
which the phase difference is changed by passage through the 
compensator. Hence, 



27rA - - (13-9) 

The positions of the axes of the incident polarized light may 
also be found. To do this let plane-polarized light fall on the 
compensator and move wedge B through a distance s/2, having 
previously calibrated the micrometer driving the wedge in terms 
of the distance 2s between the dark fringes. Then the cross 
hairs will be over a position at which the phase difference is 7r/2, 
corresponding to a retardation of A/4. Now let the elliptically 
polarized light fall once more on the compensator. In general 
the middle black band will not be under the cross hairs, but it may 
be brought there by rotating the compensator. It will usually 
be necessary to rotate the analyzer also to obtain maximum 
distinctness of the fringes. The axes of the incident elliptically 

polarized light are now parallel to the 

^w 

axes of the wedges. 

The situation is now as shown in 
Fig. 13-21. OA and OB are parallel 
to the axes of the two wedges, OC is 
the direction of the principal section 
of the analyzer, and the direction of 
vibration of the light which is extin- 
guished at the central fringe ia,DD'. 
If the analyzer is rotated through the 




F ^ 13 21 angle 0, the fringes will disappear, 

since for this position the compen- 
sator will act like a quarter-wave plate. The tangent of will 
be the ratio of the axes of the incident elliptical polarization. In 
the illustration the longer axis is in the direction OA. 

13. The Reflection of Polarized Light. The electromagnetic 
theory of light tells us that if a plane wave is incident upon the 
boundary between two media, the character of the reflected and 



SBC. 13-13] 



POLARIZATION OF LIGHT 



233 



refracted waves will depend upon the state of polarization of the 
wave as well as upon the character of the two media and the angle 
of incidence. Consider a vector (Fig. 13-22) representing an 
electric force at an angle with the plane of incidence to be resolved 
into components of amplitude a and 6 perpendicular and parallel, 
respectively, to that plane. Then for an isotropic transparent 
medium the components of amplitude 
ai and 61 in the reflected wave are 
shown to be given by 



sin (i r) 

CL\ = a~ j-. j r> 

sm (i + r) 



(13-10) 



- h 

l = 



tan *' ~ 





Surface 




tan (i + r) 

These equations were originally 
derived by Fresnel for the transmis- 
sion of light on the assumption of an 
elastic-solid medium, although certain 
of his assumptions have not been able 
to withstand the test of experiment. Flo> 13 ' 22 - 

The equations themselves, however, have been experimentally 
proved correct. 

An examination of eqs. 13-10 and 13-11 discloses that no matter 
what i may be, a\ never becomes zero, while for i + r = 90, 
61 0. This corresponds to the condition for maximum polariza- 
tion, in agreement with Brewsler's law. 

If the second medium has a higher index of refraction than 
the first, i > r, and by eq. 13-10, ai and a are of opposite sign, 
which can be interpreted as meaning that on reflection there is a 
change of phase IT in the vibrations perpendicular to the plane of 
incidence. The vibrations 61 and b parallel to the plane of 
incidence are alike in sign if i + r < 90, and different in sign 
if i + r > 90. In Fig. 13-22, where i + r is taken less than 
90 deg. this sign convention, which may also be interpreted as a 
change of phase, is illustrated. 

For normal incidence, the sine and the tangent may be replaced 
by the angle and, in the limit, eqs. 13-10 and 13-11 become, on 
combining with i = nr (SnelFs law) 

a, . -<A^4> (13-12) 

n -f- 1 



234 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIII 

and . 

*>-if (13-13) 

Since a change of phase occurs on reflection, it follows that for 
the case of normal incidence, provided n > 1, there will be a 
node at the surface and standing waves set up if the reflection 
takes place at the surface of a denser medium. Actually the 
node will be at the surface only if n = <x> , i.e., if the reflected and 
incident waves are equal in amplitude; for reflection at ordinary 
media there is only a minimum at the surface. From the 
postulates of the electromagnetic theory it follows also that in 
the equation similar to eq. 13-13 for the magnetic force, like signs 
in the two sides really mean no difference in phase in the incident 
and reflected amplitudes. In order to determine whether the 
light is the electric force or the magnetic force in an electro- 



V///77 




'- Mirror 



Fio. 13-23. Fringes appear OH the film where it intersects loop. 

magnetic wave, Wiener, Drude, and others performed experi- 
ments in which the standing waves were recorded on extremely 
thin transparent photographic or fluorescent films on plates 
inclined at small angles to a mirror surface, as illustrated in Fig. 
13-23. These experiments proved conclusively that what we 
have described as transverse light vibrations consist of the electric 
disturbance in an electromagnetic wave of light frequency. The 
experiments also showed that the node at reflection was not at 
the surface, but a very small distance below it. 

14. Rotation of the Plane of Vibration on Reflection. If the 
light incident on a surface is a plane-polarized beam of amplitude 
A vibrating in a plane making an angle a with the plane of inci- 
dence, the vibration can be resolved into two components, one 
of amplitude a = A sin a perpendicular to, and one of ampli- 
tude b = A cos a parallel to, the plane of incidence. For a 
transparent isotropic medium the reflected light will thus have 
components of amplitude 

A . sin (i r) / ^\ 

ai = A sin a. - ;.-, ( (13-14) 

sin (i H- r) 



SEC. 13-15] POLARIZATION OF LIGHT 235 

perpendicular to, and 



61 = A cos a an V* r > (13-15) 

tan (i + r) v ' 

parallel to, the plane of incidence. The reflected plane-polarized 
vibration will in general lie in a plane inclined to the plane of 
incidence at an angle 0, and the components of the reflected beam 
may therefore also be written 

a x = B sin 0, (13-16) 

6, = B cos 0, (13-17) 

perpendicular and parallel, respectively, to the plane of incidence. 
From eqs. 13-14, 13-15, 13-16, and 13-17 it follows that 

-tan a ^ = tan 0. (13-18) 

cos (i + r) v ' 

When the light is incident normally, then tan a = tan 0. As 
the angle of incidence increases, /? becomes greater than a until, 
when the angle of complete polarization is reached, tan /3 = oo, 
and = 90, in accordance with Brewster's law. For greater 
angles of incidence, /? > 90, (90 /3) becomes negative and 
finally equals a when i = 90. Equations analogous to those 
above may be developed for the case of refraction, and for the 
reflection and refraction of elliptically polarized light. The 
conclusions thus reached have been tested by experiment and 
found to be valid. 

15. The Nature of Unpolarized Light. The descriptive 
mechanism employed in the discussion of polarized-light phe- 
nomena often leads the student to infer that the nature of 
ordinary light, which has suffered no reflection from, nor trans- 
mission through, material media, is disclosed by a dogged applica- 
tion of the mechanical picture of linear vibration components. 
A typical question which arises is, "What is the form of the 
transverse vibration in ordinary (unpolarized) light in, say, some 
unit element of the beam?" It seems worth while to clarify 
this point by indicating the limits to the use of the descriptive 
mechanism of the phenomena of polarization. In order to 
explain these phenomena, it is found convenient to consider 
separately the components of transverse vibration perpendicular 
and parallel to a given plane. In many cases, experiment 



236 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIII 



j 1 



proves that the transverse light vibration is actually decomposed 
by a medium into two such components; an illustration is the 
birefringence of ordinary calcite. The error occurs if an attempt 
is made to extend this form of representation to explain the 
nature of the original unpolarized beam. Thus we find that 
either by implication or explicit statement, ordinary (unpolar- 
ized) light from a source is sometimes spoken of as being actually 

made up of innumerable transverse 
linear vibrations with all possible ori- 
entations in a plane perpendicular 
to the direction of propagation. 
There is not the slightest experi- 
mental evidence for this point of 
view. 1 Such evidence tells us only 
that if ordinary (unpolarized) light 
is broken up into plane components 
by some sort of polarizing device, the 
amplitudes in all orientations in the 
original beam are shown to be equal. 2 
16. The Fresnel Rhomb. From a 
consideration of the change in phase 
suffered by light reflected inside iso- 
tropic media such as glass, Fresnel 
concluded that when plane-polarized 
-light is totally reflected internally in 
ordinary glass the components under- 
go a relative phase change of x/4. The rhomb constructed by him, 
and shown in Fig. 13-24, provides for two such internal reflec- 
tions, thus introducing a phase change of twice 7r/4, or 7r/2 
between the two components, as does a quarter-wave plate. 3 
Plane-polarized light incident upon the rhomb with its plane of 
vibration at 45 deg. to the plane of incidence will emerge as 

1 There is evidence of the existence of plane-polarized light of particular 
frequency in the Zeeman effect (see Chap. XVI), but this is another matter. 

8 An interesting experiment dealing with this question has recently been 
performed by Langsdorf and Du Bridge, Jour. Optical Soc. Amer., 24, 1, 
1934. See also subsequent comments by R. W. Wood, Jour. Optical Soc. 
Amer., 24, 4, 1934, R. T. Birge, Jour. Optical Soc. Amer., 26, 179, 1935, and 
L. DuBridge, Jour. Optical Soc. Amer., 25, 182, 1935. 

*"For a detailed description of the Fresnel rhomb see Drude's "Theory of 
Optics." 




Fia. 13-24. Polarization by 
Fresnel rhomb. 



SEC. 13-17] POLARIZATION OF LIGHT 237 

ircularly polarized light. Moreover, the effect is the same for 
ill wave-lengths, since the variation of index of refraction with 
wave-length is insufficient to cause any trouble. The disadvan- 
tage of the Fresnei rhomb is that the emergent beam, while in 
the same direction as the incident, is displaced sideways with 
respect to it so that a rotation of the rhomb causes a movement 
of the image which is difficult to follow with other apparatus. 

17. General Treatment of Double Refraction. Thus far 
the subject of double refraction has been confined to uniaxial 
crystals. These are characterized by possessing a single direc- 
tion through the crystal for which there is a common tangent to 
the wave fronts of two vibrations in planes perpendicular to each 
other. In biaxial crystals, however, the mechanism of wave 
propagation is not so simple. In order to explain in a most 
general way the optical properties of transparent crystalline 
media, Fresnei developed a theory which, though founded on 
assumptions which may be criticized, 
gave an accurate representation of the 
experimental facts. The intention here 
is simply to give the conclusions reached 
by the theory as to the form of the light 
waves propagated through crystals, and 
their state of polarization. 1 We may 
suppose that at any instant many plane 
waves are traveling in different direc- 
tions through a point (Fig. 13-25) in 

a crystal. For each such plane wave 




there will be, in general, directions of surface is given by the curved 
maximum and minimum velocities of 

propagation at right angles to each other. The form of the 
wave surface after a given time is represented by the curved line 
in Fig. 13-25; it is the common tangent to the plane waves at 
that instant. Since in any direction there are in general two 
wave velocities, this wave surface consists of two surfaces or 
sheets, only one being shown in the figure. The equation of the 
wave surface derived by Fresnei is 

1 For an extended treatment of Fresnel's theory see Preston, "Theory of 
Light"; or Schuster, "Theory of Optics." For a complete treatment of the 
subject from the standpoint of the electromagnetic theory of light see 
Born, "Optik." 



238 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIII 

-T^l? + -^T5 + -T-^2 = 0, (13-19) 

r 2 a 2 r 2 6 2 r 2 c 2 

where a, 6, and c represent the velocities of propagation through 
the crystal of the vibrations which are respectively parallel to 
the x-, y-j and z-axes, and r 2 = z 2 -f- y 2 + z 2 - We may consider 
that a > b > c. Equation 13-19 may be written 

r' 2 (a 2 * 2 4- by 2 + cV) - a' 2 (6 2 + r 2 )* 2 - 6 2 (c 2 + a 2 )?/ 2 

- c 2 (a 2 + 6 2 )z 2 + a 2 6V 2 = 0. 

The properties of this surface are more easily examined by 
studying its projections on the three planes, each of which is 
defined by a pair of the coordinate axes. 

By putting x = 0, we obtain the intersection of the wave 
surface and the t/z-plane. The equation of this intersection is 

(r 2 - a 2 ) (6V + c 2 * 2 - 6V 2 ) = 0, 
since in the ?/z-planc r 2 = y~ -f- z' 2 , and is satisfied by 

y 2 + z 2 = a' 2 , 
a circle of radius a, and 

+ - - 1 
c 2 ^ 6 2 ~ ' 

an ellipse with semiaxes 6 and c, lying entirely within the circle. 
Similarly, putting 2 = 0, the intersection with the xy-plane is 
shown to be 



a circle of radius c, and 



! -i- HI = 

6 2 "*" a 2 



an ellipse with semiaxes a and 6, lying outside the circle. The 
intersection with the zz-plane, obtained by putting y = 0, is 



a circle of radius 6, and 



__ |_ _ _ 1 

c 2 ^ a 2 ' 



SEC. 13-17] 



POLARIZATION OF LIGHT 



239 



an ellipse with semiaxes a and c which is cut by the circle at 
four points. 

These sections are illustrated in Figs. 13-26a, b, and c. The 
form of the entire wave surface is illustrated in Fig. 13-27. The 
differences between the velocities a, b, and c are exaggerated. 

Z Z Y 





the 



(a) (b) 

FIG. 13-26. Intersections of the wave-surface by: (a) the z/2-plane; (6) 

a!-plane; (c) the xy-plane. 

In accordance with the fundamental assumptions upon which 
the theory is based, a plane-polarized wave in which the vibra- 
tions are parallel to the ^-direction is thought of as traveling 
through the crystal in any direction with the velocity *tt. For 
such waves the index of refraction is V/a, where V is the velocity 
of light in a vacuum. Similarly, the 
indices for plane-polarized waves 
in which the vibrations are parallel 
to the y- and z-directions, respec- 
tively, are V/b and V/c. These 
three ratios, V/a, V/b, V/c, are 
called the principal indices of refrac- 
tion of the crystal. The use of Fig. 
13-27 will enable the reader to 
understand more precisely the man- 
ner in which the velocities corre- 
spond to the directions of vibration. 
It shows the intersections of the 
wave surfaces with the three coor- 
dinate planes in one quadrant. Consider the zz-plane : Vibrations 
perpendicular to it, i.e., parallel to y, have velocity 6, no matter 
in which direction they travel through the crystal; vibrations in 
the zz-plane parallel to the ^-direction travel in the z-direction 
with velocity a, and vibrations parallel to the z-direction travel in 
the z-direction with velocity c. Vibrations oriented otherwise 
in the plane do not have velocities intermediate between a and c, 




FIG. 13-27. 



240 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIII 



but are transmitted as though decomposed into component vibra- 
tions parallel to the x- and 2-directions which travel with the 
velocities a and c, respectively. 

The transmission velocities of vibrations parallel and perpen- 
dicular to the two other coordinate planes may be obtained from 
the figure in the same way. The double-headed arrows represent 
the directions of vibration. 

It is to be noted that the ray through the crystal is not in 
general normal to the corresponding wave front but is the line 
from the point of incidence drawn to the point of contact of the 
tangent plane to the wave surface. 

18. Optic Axes in Crystals. There are two directions in a 
crystal along which plane waves may be transmitted with a 
single wave velocity, no matter what the directions of vibration 
in their wave fronts may be. These are called the optic axes. 
In Fig. 13-28 they are OM and OM', since tangents to the circle 
at M and M' are also tangent to the ellipse at N and N'. That^is, 
a wave front which after refraction is parallel to MN (or M'N') is 

propagated as a single wave in 
the direction OM (or O'M'), 
whatever the direction of the 
vibrations in the wave front 
may be, but the rays correspond- 
ing to the vibrations with different 
orientations have different direc- 
tions. For instance, for waves 
vibrating in the zz-plane the ray 
lies in the direction ON, while 
for waves vibrating perpendicu- 
larly to the zz-plane and in the 
tangent plane the ray has the 
direction OM. The tangent 
plane, however, which inter- 
sects the plane of incidence in 
M N, touches the wave surface in a ring, with the point P in the 
middle of a slight depression in the surface. 

Consequently, if a narrow bundle of ordinary light is incident 
on a section of a biaxial crystal so that after refraction the wave 
normal proceeds along the optic axis, a single ray may have any 
one of the infinite number of directions represented by a line 




FIG. 13-28. 



SEC. 13-19] 



POLARIZATION OF LIGHT 



241 



from to a point on the circle of contact, depending on the 
particular direction of its vibration, and the rays of the entire 
bundle spread out so that each ray becomes a line in the surface 
of the circular cone whose apex is at O and whose base is the 
circle MN. This is shown by the phenomenon of internal conical 



Screen 





Fio. 13-29. Internal conical refraction. 

refraction. A crystal C (Fig. 13-29) is cut with its faces per- 
pendicular to the bisector of the angle between the optic axes, 
and a narrow pencil of light is allowed to fall on a limited area 
of the surface. In general there will be two images of the hole 
on the screen, but for a certain direction O'O of the incident beam 
there is a ring of light which has the same diameter for different 
distances from the screen. The angle of the cone 
of refraction agrees with that predicted by theory. 
19. Axes of Single Ray Velocity. The two 
directions OP and OP' are called the axes of single 
ray velocity. At the points P and P' there is an 
infinity of tangents to the surface, two of which, in 
the plane of incidence, are illustrated in Fig. 
13-30. This is another way of saying that there 
is an infinity of wave normals at P. Since the 
direction of emergence of the light into the air depends upon 
the direction of the wave normal in the crystal, there will be, 
corresponding to a ray traversing the crystal in the direction 
OP (or OP') a hollow cone of rays leaving the crystal. This 
phenomenon, called external conical refraction^ can be demon- 
strated by the use of the same crystal that is used for demonstrat 




Fiti. 13-30. 



242 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIII 




ing internal conical refraction. A convergent beam of light is 
focused on a small hole in a screen covering one surface, as in 
Fig. 13-31. In a screen over the other surface is another hole 0', 
for which a position may be found so that a hollow cone of rays 
is refracted into the air. The direction 00' is the axis of single 
ray velocity. Actually, of course, only the light in a similar 
hollow cone on the incident side is thus refracted through 0'. 

The rest of the light which 
passes through is refracted in 
other directions and is stopped 
by the screen. 

The meaning of the phrase 
" single ray velocity" is made 
clear by this experiment, for it 
is obvious that while the ray 
00' (or, rather, a narrow bun- 
die of rays) is made up of waves 
vibrating in any plane what- 
ever, these all have the same 
velocity through the crystal. 

Lens Thus along the directions OP 

and OP 1 (Fig. 13-28) is fulfilled 

Fu, 13-31. -External conical refraction. the C dition that a difference 

in optical length between two 

points on the ray is independent of the plane of vibration. 
The relation between this characteristic in biaxial and uniaxial 
crystals is now apparent, for the optic axis in a uniaxial crystal is 
that direction through the crystal for which the condition just 
stated is fulfilled. The mechanics of wave propagation in uniaxial 
crystals is thus seen to be a special case of the more general 
mechanics of wave propagation in biaxial crystals, and is, in 
effect, that case for which the ellipse of Fig. 13-28 is tangent to 
the circle. If the ellipse is inside the circle and tangent to it at 
two points, the wave surface represented is that of a positive 
uniaxial crystal; if the ellipse is outside the circle and tangent to 
it at two points, the wave surface represented is that of a negative 
uniaxial crystal. In either case, the line through the crystal 
connecting the points of tangency is the optic axis of the crystal. 
20. Rotatory Polarization. If a pair of nicols is crossed so as 
to extinguish the incident light, an ordinary isotropic substance 




SBC. 13-30] POLARIZATION OF LIGHT 243 

placed between them will produce no effect. The same thing 
is true if a thin section of calcite with its faces perpendicular to 
the optic axis is placed between the nicols, provided the light is a 
parallel beam. But if a thin section of crystal quartz so cut is 
used, light will be transmitted by the analyzer. The analyzer 
may then be rotated to an angle at which the light will once more 
be extinguished, proving that after passing through the Quartz 
plate, it is still plane-polarized but vibrating in a plane different 
from that of the light incident on the quartz. From the ordinary 
laws of double refraction the failure of the calcite to produce any 
effect was to have been expected, since both the ordinary and 
extraordinary rays traverse the crystal in the direction of the 
optic axis with the same velocity, and no difference of phase is 
introduced between them. From these considerations also the 
same result might have been expected with the quartz plate, but 
instead there is definite evidence that a rotation of the plane of 
vibration has taken place. The use of thicker plates of quartz 
will show that the angle of this rotation is proportional to the 
thickness traversed. 

Crystal quartz and other substances which have this power to 
turn through an angle the plane of vibration of a polarized beam 
transmitted along the optic axis are said to be optically active. 
This property is quite distinct from that possessed by half-wave 
plates in effecting a change in the plane of vibration by a relative 
retardation between the ordinary and extraordinary plane vibra- 
tions, for, as has been seen, the result may be produced when the 
light traverses the crystal in a direction along which these two 
vibrations have the same velocity. Some crystals rotate the 
plane of vibration in a right-handed (clockwise) direction and 
others in a left-handed (counterclockwise) direction, and are 
accordingly called right-handed or dextrorotatory and left- 
handed or levorotatory. Quartz occurs in both forms, the 
crystal symmetry of one being the mirror image of that of the 
other. A rotation is said to be right-handed when the observer 
looking toward the light source sees the plane of polarization 
rotated in a clockwise direction. If the crystal is turned around 
so that the light traverses it jn the opposite direction, no change 
in the direction of rotation of the plane of polarization is observed; 
destruction of the crystal state, as in the case of fused quartz, 
destroys the optical activity of the substance. 



244 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIII 

Many liquids and organic substances in solution have been 
found to be optically active. A solution of an active substance 
in an inactive one possesses a power of optical rotation which is 
proportional to the amount of active substance present in a given 
quantity of solution. While this rule is generally true, in some 
cases the rotation is found to vary slightly with the nature of the 
solvent. An approximate formula for the rotation p, in degrees, 
is given by 

p = A + Bs + Cs 2 , (13-20) 

where s represents the weight of the solvent in 100 parts by 
weight of solution, and B and C are empirical constants. A is 
defined as the molecular rotation of the pure substance, for s = 0, 
molecular rotation in turn being defined as the amount of rota- 
tion of the plane of polarization produced by a column 10 cm. in 
length containing 1 gm. of the substance per cubic centimeter, 
or, as the amount of rotation produced by a thickness of 10 cm. 
of the pure substance divided by the density of the substance. 
For most purposes the second two terms on the right-hand side 
of the equation are negligible. The molecular rotation, some- 
times called the specific rotation, of sucrose (cane sugar) for the 
D-line of sodium (5893 angstroms) is +66.67 deg., the positive 
sign indicating that the substance is right-handed. The almost 
complete dependence of the angle of rotation on the density of 
the optically active substance in solution has made the rotatory 
power an extremely useful means of determining the purity of 
sugar. The effect produced by a given thickness of a particular 
sample may be compared with that of one of standard purity, 
and the percentage of foreign substance may thus be determined. 

The angle of rotation of the plane of vibration in optically 
active substances is nearly proportional to the reciprocal of the 
wave-length squared. If the proportionality were exact, the 
law could be written p = K/\ z , where K is a constant. A better 
agreement with experiment is obtained if to the right-hand side 
of the equation are added terms whose values depend on natural 
free periods of vibration in the crystal. l The relation between the 
rotation and wave-length is called dispersion of the rotation. 

21. FresnePs Theory of Rotatory Polarization. Fresnel 
assumed that the incident plane-polarized light upon entering the 

1 For a more complete discussion of this topic, see Drude, "Optics." 



SBC. 13-21] 



POLARIZATION OF LIGHT 



245 



quartz was broken into two beams circularly polarized in opposite 
directions and propagated through the crystal with different 
velocities. In Fig. 13-32a and 6 are represented two opposite 
circular vibrations in which, when they are superposed, the linear 
components in the X-direction will neutralize each other, leaving 
only a plane vibration in the F-direction as in Fig. 13-32c. But 
if the left-handed vibration travels through the crystal faster 




(a) 



(b) (c) 

Fi. 13-32. 



(d) 



than the right-handed, after emergence from the crystal the 
components in a direction represented by U (Fig. 13-32rf) will 
neutralize each other, leaving only a plane vibration in the 
F-direction. The angle between the Y- and V-directions depends 
on the relative velocities of the two circular vibrations and the 
thickness of crystal traversed. 

If this explanation is correct, then the two vibrations, since 
they travel with different velocities, should undergo different 




Fio. 13-33. 

amounts of bending by refraction at an oblique surface. Fresnel 
found that the resulting separation of the beams, while very small, 
could be detected if a narrow beam is passed through a block 
made of alternate prisms of right- and left-handed quartz, as 
illustrated in Fig. 13-33, the prisms being cut so that the light 
traversed them in the general direction of the optic axis. While 
in the first prism the left-handed circularly polarized beam travels 
faster than the right-handed, in the second it travels slower. 
Since the light is incident upon each oblique face at a large angle, 
refraction of the two components takes place at slightly different 
angles at each face, thus increasing the separation of the beams. 



246 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIII 



Upon examination with a quarter-wave plate and an analyzer, 
the two beams may be shown to be circularly polarized in opposite 
directions. 

The rotation of the plane of vibration can be described analyti- 
cally as follows: Let 



X\ a 



d 



= a sin - - 



represent a left-handed circular vibration, and 



d 
= a cos - - - 



T r wi i (t 

i z (i sin -7-1 t 



) 



a right-handed circular vibration, the two having the same period 
and amplitude and traveling in the same direction with velocities 
V} and v z , respectively, through a crystal of thickness d. Super- 
position of the two can be represented by adding the X-compo- 
nents and the F-components separately, i.e., 



X = X, 

Y = yi 



F 27r A 
= a[ cos ^^ - 






cos 



y, = a 



. 27T 

sm ~ 



. 27T/ r/\l 

~ sm "^ v s;} 



which may be changed to the form 



X = 2a cos -=f 
y = 2a cos -7=7 



* 2U, + i 



4 I I * 

1 2' + 



Trrffl 1 1 
cos -7=7 > 

^1^2 VlJ 



(13-21) 



according to which the X- and y-components of the combined 
vibration have the same phase, hence the result is a plane- 
polarized vibration. The plane of this vibration is given by 



(13-22) 



y _ T^/i i 

x ~ tan Y^ ~ r 



SEC. 13-22] 



POLARIZATION OF LIGHT 



247 



which varies with d, the thickness of the crystal traversed. The 
angle of rotation corresponding to any thickness d is therefore 



(13-23) 




p--srt--- 



22. The Cornu Double Prism. Since crystal quartz transmits 
ultraviolet radiations of wave-length as short as about 1800 
angstroms, it is extremely useful in the construction of lenses 
and prisms for spectrographs. But in the preceding section it was 
shown that ordinary light incident obliquely upon the surface of 
a quartz prism suffers double refrac- 
tion, even if the light traverses the 
prism in the direction of the optic 
axis. This not only results in 
double images in a single quartz 
prism, but the rotation of the plane 
of polarization is a disadvantage in 
spectroscopic observations in which 
the measurement of polarization is 
involved. The Cornu prism, designed to eliminate these double 
images, is constructed of two 30-deg. prisms, one of right- 
handed, the other of left-handed quartz, cut so that the light 
travels in the direction of the optic axis in each. The two are 
placed together, as illustrated in Fig. 13-34. The amount of 
rotation in the first prism is exactly neutralized by the rotation 
in the opposite direction in the other. This arrangement is 




FIG. 13-34.- 



The Cornu double 
prism. 



tMirror 




t 
\i 



__- -- - - ^ -- ~ ~ ___ ^Spectrum 



Fiu. 13-35. The Littrow mounting. 

unnecessary if the spectrograph is of the Littrow type, illustrated 
in Fig. 13-35. In this instrument a single lens serves for both 
collimator and camera, and the light is reflected back through 
the prism from a coat of metal deposited on its rear face. Since 
for either right- or left-handed quartz the angle of rotation is the 
same no matter in which direction the light travels along the 
axis, the rotations produced in the incident and reflected paths 
neutralize each other. 



248 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIII 

23. Half -shade Plates and Prisms. Measurements of rotatory 
polarization may be made by setting a pair of nicols for extinction, 
interposing the optically active substance between them, and 
recording the angle through which the analyzer is turned to 
produce extinction once more. This method is not very accurate, 
because it is difficult to tell just when the light is completely cut 
off. Since a determination of equality of intensity of two parts 
of the field may be made with greater accuracy, it is customary to 
introduce into the optical path some device for substituting this 
setting for the setting for extinction. One of these, the Laurent 
half-shade plate, consists of a semicircular half-wave plate of 
quartz or other crystal set between the polarizer and analyzer 
and close to the former, with its optic axis at a small angle 6 
with the principal section of the polarizer. In order to com- 
pensate for the absorption and reflection of this plate, the other 
half of the field is covered with a piece of glass of appropriate 
color and thickness. The smaller the angle 0, the greater the 
change in relative intensity of the two halves of the field of view 
when the analyzer is rotated. For small values of 6, however, 
the intensity in both halves will be small. For this reason it is 
customary to mount the half-wave plate to permit its adjustment 
over a small range of angle. The observations are made by 
turning the analyzer until the two halves of the field are equally 
bright. In some instruments a small nicol covering one half of 
the field is substituted for the Laurent plate. 

Another device which serves the same purpose as the Laurent 
plate is the Cornu-Jellet prism. This is made by splitting a 

nicol in a plane parallel to the 
direction of vibration of the trans- 
mitted light, and removing a sec- 
tion, as illustrated in Fig. 13-36a. 
(a) (b) When the two pieces are joined 

FIG. 13-36. The Cornu-Jellet together, as showri in Fig. 13-366, 

the planes of vibration of the light 

transmitted by the two halves make a small angle with each other, 
and extinction takes place for different settings of the analyzer. 
When the beam through one half is extinguished, a small amount 
of light is transmitted through the other. In making observa- 
tions of the rotation of the plane of polarization in a substance, 
it is customary to set the analyzer with its plane of transmission 





SEC. 13-23] POLARIZATION OF LIGHT 249 

so that the two halves are equally bright. When the plane of 
rotation of the incident light is rotated through a given angle 
by the substance under examination, the analyzer may be set 
with great accuracy to the new angle at which the two halves 
of the field once more appear equally bright. 

Problems 

1. How thick must a quarter- wave plate be if it is made of quartz? 
In what direction must its faces lie with regard to the optic axis? 

2. What is the refractive index of a piece of glass if the light of the 
green mercury line (5461 angstroms) is plane-polarized when reflection 
is at an angle of 5747'? 

3. A Wollaston prism is made of quartz, each prism having an angle 
of 45 deg. If it is used so that the incident light is normal to the surface 
of the prism, what will be the angle between the two emerging beams? 

4. A plate of quartz 0.54 mm. thick is cut with its faces parallel to 
the axis. If a beam of plane-polarized light of wave-length 5461 
angstroms is incident normally on the plate, what will be the phase 
difference between the two emergent beams? 

5. If the direction of vibration of the incident plane-polarized beam of 
Prob. 4 makes an angle of 30 deg. with the optic axis of the quartz, 
what will be the character of the polarization of the emergent beam? 
Give full details. 

6. A solution of camphor in alcohol in a tube 20 cm. long is found to 
effect a rotation of the plane of vibration of the light passing through 
it of 33 deg. What must be the density of the camphor in grams per 
cubic centimeter in solution? The specific rotation of camphor is 
+54 deg. at 20C. 

7. Consider the experiments described in Sees. 13-ld and e. A 
slight tilting of the crystal in either case will reveal that in light trans- 
mission along the axis the two dots appear to be at different depths 
in the crystal, while in transmission perpendicular to the axis they 
appear almost at the same depth. This is contrary to theory. Explain 
the apparent contradiction, considering the light to be divergent. 



CHAPTER XIV 

SPECTRA 

1. Kinds of Spectra. In general there are three kinds of 
spectra : 

a. Bright Line Spectra. These have their origin in incandescent 
gases at low pressure as in a partly evacuated discharge tube, in 
flames, in the glowing gas between the terminals of an electric 
arc or spark, and in certain so-called gaseous nebulae such as the 
irregular nebula in Orion, and in the tails of comets. The 
spectrum seen when only the edge of the sun is observed through 
a spectroscope is a bright line spectrum. In the majority of 
cases, bright line spectra are those of monatomic gases, although 
certain of them called band spectra are due to molecules. 

6. Continuous Spectra. These are due to incandescent solijds 
or liquids, such as a lamp filament, the poles of an electric arc, 
molten metals of high melting point, and also to incandescent 
gases at high pressure such as exist in lower levels in stars. 

Gases at low pressures for which the most conspicuous spectrum 
is one of bright line emission may under certain circumstances 
also emit a spectrum which is continuous over a given spectral 
range. The manner in which spectral theory explains this type 
of emission will be given in later sections. 

c. Absorption Spectra. These are in general of two sorts: 
continuous absorption, and line absorption. A type of con- 
tinuous absorption, i.e., over a considerable range of wave-length, 
will be discussed briefly in the sections on dispersion. Line- 
absorption spectra commonly occur when the light from a source 
emitting a continuous spectrum is observed with a spectroscope 
after passing through gases at low pressure and lower temperature 
than that of the source. For the production of this type of 
spectrum, it is necessary that the atoms or molecules of the 
intervening gas be in a condition. to absorb energy of radiation 
which strikes them. The mechanism of this line absorption is 
explained on the basis of the quantum theory of spectra. 

2. Early Work on Spectra. Although the dispersion of light 
into a spectrum by a prism had been studied by Kepler and others, 

250 



SEC. 14-3] SPECTRA 251 

Newton was the first to formulate the precise laws of dispersion. 
He invented the word " spectrum" for the band of color obtained 
from the sun's light with a prism. Fraunhofer's discovery of the 
significance of the absorption lines in the solar spectrum 1 was 
the beginning of what might be called the first pioneering era in 
spectroscopy, which lasted for about 30 years after Fraunhofer's 
discovery. 

The relationship existing between laboratory and celestial 
spectra was first clearly stated by Kirchhoff in 1859. By 
exhaustive experiments carried out in collaboration with Bunsen, 
he showed that if a burning salt is placed between any hotter 
source of a continuous white-light spectrum and the spectroscope, 
there will be seen a spectrum crossed by absorption lines whose 
positions coincide with the bright lines obtained from the burning 
salts alone. His conclusion was that the cooler flame absorbs 
light of the same wave-length as it will emit. He inferred there- 
fore that the Fraunhofer lines in the solar spectrum are due to 
the presence of a solar atmosphere cooler than the underlying 
body of the sun, and containing the same elements which give rise 
to corresponding bright lines in the laboratory. A year or two after 
the announcement of these conclusions he published the funda- 
mental law of radiation and absorption: The ratio between the 
absorptivity and emissive power is the same for each kind of rays 
for all bodies at the same temperature. 

3. The Balmer Formula for Hydrogen. The half century 
following Kirchhoff was a period of accelerated accumulation of 
experimental data and technique, much of which was in the field 
of astrophysics. Engrossing as the story of these developments 
may be, it cannot be told here. 2 The first step toward precise 
knowledge of the origin of spectra was made in 1885 by Balmer, 
who showed that with a high degree of approximation the wave- 
lengths of the hydrogen lines could be fitted into the formula 



= 3645 



1 See Sec. 6-14. 

2 The reader is referred to Crew, "Rise of Modern Physics," Scheiner, 
"Astrophysical Spectroscopy," and Lockyer, "Inorganic Evolution as 
Studied by Spectrum Analysis," for the history and background of spec- 
troscopy during this period. 



252 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIV 

where n is an integer equal to or greater than 3, and X is in 
angstroms. 

4. The Rydberg Number. It is a little easier to connect this 
formula with developments to be described later if a change is 
made from wave-length (X) to wave number (v), the number of 
waves in a centimeter in vacuo. We then have 

'-^- 3\ (14-2) 





in which n = 3, 4, 5 . . . , and R has the approximate value 
109,700. In 1890, Rydberg discovered that the number R is 
with slight variations common to all elements. For any particu- 
lar element the formula is 





For instance, with this formula the wave numbers of certain 
lines in the helium spectrum are given with the atomic number 1 of 
helium Z 2, rii = 3, n 2 = 4, 5, 6, etc. It may be perplexing 
to the student that in illustrating eq. 14-3 by choosing n\ = 3 
instead of 2, for helium, nothing is said to indicate the reason for 
that choice. That reason will appear in the following sections. 

5. Series in Spectra. The lines of the hydrogen spectrum 
given by the Balmer formula (eq. 14-1) constitute what is called 
a series. Several years before Balmer's discovery it was noticed 
that in the spectra of several elements, notably those of sodium 
and potassium, most of the lines could be fitted into series. In 
these spectra there is not a single series of lines, as in hydrogen, 
but several series, each of which is extended throughout the 
spectrum. Also, the series themselves consist of doublets, 
triplets, or higher multiple groups. Since the lines in different 
series for a given element are different in appearance, some being 
predominantly strong, others diffuse, others sharp, etc., it became 
customary to classify series as principal, sharp, diffuse, funda- 
mental, etc., the letters P, S, D, F, etc., being used for convenience. 
Because of the existence of a real physical significance to these 
differences of appearance, the letters have remained in the nota- 

1 By "atomic number" is meant the ordinal number of the atom in the 
periodic table of the elements, beginning with hydrogen as number one. 



SBC. 14-6| 



SPECTRA 



253 



tion of spectral theory. In Fig. 14-1 are some reproductions of 
spectra showing series. 1 

6. The Hydrogen Series. It occurred independently to two 
workers in this field, Rydberg and Ritz, that in the Balmer 
formula (eq. 14-3) n\ might be a running integer as well as n*. 
For instance, if, as in the Balmer formula, n\ = 2, n 2 = 3, 4, 



1 1 




i 






I 






I 1 




i 












1 1 




I 






1 






1 1 


PRINCIPAL 


i 






1 






1 1 




i 






i 






1 
















1 


1" 


Ii i 
















i 1 






:| 










| 




i 


! 







DIFFUSE 



I III 



FIQ. 14-1. The spectrum of sodium, showing the principal series doublet and 
some members of the sharp and diffuse series. The principal series doublet 
(5890 and 5896 angstroms) is overexposed and surrounded by Rowland ghosts 
due to imperfections in the diffraction grating used. The other lines in the 
spectrum are due to impurities in the source. 

5, . . . , etc., the series of wave-lengths represented is 6563, 
4861, 4340 angstroms, . . . , etc., while if n\ is put equal to 1, 
and tt2 put equal to 2, 3, 4, 5, etc., successively, the calculations 
yield the wave-lengths 1216, 1026, 972 angstroms, . . . , etc., 
fractional parts of angstroms being omitted. These lines, lying 
in the far ultraviolet, were observed by Lyman. Thus far, five 
series of hydrogen spectra have been observed. These are listed 
below, designated in accordance with custom by the names of 
the original observers. The values of ni and n z are given in 
each case. 

HYDROGEN SERIES OP ATOMIC SPECTRA 
Lyman series: 



p - ^*2 j, n 2, 3, 4, 5, etc. 



1 For summary of the work in this subject see Fowler, "Report on series 
in Line Spectra," Fleetway Press, 1922; or White, "Introduction to Atomic 
Spectra," McGraw-Hill. 



254 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIV 

Balmer series: 

/I 1\ 

v = R( 22 ~ ^2 ), n = 3, 4, 5, 6, etc. 



Paschen series: 



/I _ 1\ 
= #(32 n 2 ), n = 4, 5, 6, 7, etc. 



Bracket! series: 



/I 1\ 
= #(42 ^2 1, n = 5, 6, 7, 8, etc. 



Pfund series: 





= 6, 7, 8, 9, etc. 

The last three of these series lie in the infrared, as can be 
verified by calculation. 

It is now evident that for any element a fair approximation to 
the wave numbers of series is given by eq. 14-3, and that the 
illustration at the end of Sec. 14-4 gives only one of the series 
of the helium spectrum^ Others will correspond to different 
values of n\ and n 2 , in accordance with the rules laid down. It 
must be kept in mind, however, that only the simplest of series 
formulas have been presented here. Others which give a closer 
approximation to the wave-lengths may be found in treatises on 
the subject of line spectra. 

7. The Quantum Theory of Spectra. The attention of many 
spectroscopists was focused upon series relations in spectra from 
the time of Balmer's discovery, but it was not until the announce- 
ment of the Bohr theory about 30 years later that the physical 
significance of these relations was disclosed. Series relations 
still are spectroscopic tools of tremendous power, for, by their 
aid, if several lines of a spectral series are experimentally identi- 
fied, the wave-lengths of the rest can be predicted. During the 
quarter century following Balmer's discovery, however, there 
remained unsolved the riddle of spectra: What, precisely, is the 
connection between the wave-lengths of the radiation and the 
changes in the atoms or molecules from which that radiation is 
emitted? The answer to this question did not follow from series 
relations in spectra. Several lines of investigation converging 
to a common end, and brought together by the insight of Niels 



SEC. 14-9] SPECTRA 255 

Bohr, gave us a theoretical explanation which later was expanded 
into what might be called the quantum theory of spectra. The 
lines of investigation contributing to this end will now be taken 
up in detail. 

8. KirchhofFs Law of Emission and Absorption. In order to 
trace the events leading to modern theories of spectra, it is 
necessary to outline the developments in the field of radiation 
which followed the work of Kirchhoff. 1 

The relation between emission and absorption deduced by 
Kirchhoff and mentioned in Sec. 14-2 rests on more extensive 
grounds than observations on solar aborption. A study of their 
characteristics shows that certain substances such as lampblack, 
deep-piled velvet, etc., absorb a greater amount of the radiation 
that falls upon them than other substances, while at the same 
time they also act as good radiators. Experiments with such 
surfaces led to the theoretical concept of a perfect black body, 
which may be defined as one whose surface absorbs all the radia- 
tion falling upon it. Obviously this (ideal) surface does not 
reflect at all. For any surface, the fraction of the radiation, 
falling upon its surface, which is absorbed is called the absorp- 
tivity of the surface. Similarly, the emissive power of a body is 
defined as the total radiation emitted per unit time per unit area 
of its surface. From both experimental and theoretical con- 
siderations, Kirchhoff was led to the general law of radiation 
connecting these quantities and stated in the next section. 

9. KirchhofPs Radiation Law. The relation between absorp- 
tivity (A) and emissive power (E) is given by Kirchhoff 's law, 
which may be stated as 

Tjl 

-j = constant (14-4) 

at a given temperature, for all bodies. Since A 1 for a so-called 
perfect black-body radiator, it follows that the law may be stated, 
in words: at a given temperature, the ratio of the emissive power to 
the absorptivity of any body is the same for all bodies and is equal 
to the emissive power of a black body at the same temperature. 

1 For a more detailed discussion of these developments, as well as of the 
quantum theory of spectra, the reader is referred to Richtmyer, "Intro- 
duction to Modern Physics," McGraw-Hill, and Reiche, "Quantum 
Theory," Dutton. 



256 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIV 



10. Stefan-Boltzmann Law. In 1879, Stefan proposed the 
relationship 

E = ff T\ (14-5) 

where E is the total emissive power of a black body at an absolute 
temperature T, and <r* is a constant. Subsequently, the same 
law was deduced theoretically by Boltzmann, who applied the 
reasoning of the Carnot cycle of energy exchange in heat engines 
to a hypothetical engine in which radiation was the working 
substance. The Stefan-Boltzmann law has been verified 

experimentally. It should be 
emphasized that the radiation dealt 
with has a continuous spectrum. 

11. Wien's Displacement Laws. 
In 1893, on the basis of classical 
thermodynamics, Wien announced 
his famous wavelength-temperature 
displacement law, which may be put 
in the form 



T = constant, (14-6) 

Wavelength 
FIG. 14-2. showing the dis- in which X mM is the wave-length for 

placement toward the violet, with which tjiere j s the max i mum energy 
increasing temperature, of the 

wave-length (x majt ) of the energy of radiation at absolute temperature 
maximum. y The yalue ^ ^ e constant prod- 

uct is found to be 0.2884 cm.-deg. By the particular nature of 
the theoretical considerations upon which the law is based, Wien 
was enabled to obtain another relation, namely, the energy- 
temperature displacement law, which may be written 



t 



UJ 




= constant X 



(14-7) 



where Em** is the energy of radiation at the maximum for T. 
These laws have been verified by experiment. Equation 14-6 is 
illustrated graphically in Fig. 14-2. Further comprehension 
of the significance of the displacement laws may be gained by 
consideration of Fig. 14-3, in which are plotted as ordinates 
experimental values of E\/T* against values of XT' as abscissae 
for three temperatures. E\ is an expression for the energy 

* The experimental value of a is usually given as 5.735 X 10~ 6 erg cm.~ a 
deg.~~ 4 sec." 1 . 



SBC. 14-12] 



SPECTRA 



257 



corresponding to a given wave-length, and may be called the 
monochromatic emissive power. As predicted by theory, tho 




FIG. 14-3. Experimental verification of the displacement laws of black-body 
radiation. (From Richtmyer, Introduction to Modern Physics.) 

curve is the same for all temperatures. From this it may be 
concluded that E\/T 5 is some unknown function of XT, or 



Combining this with eq. 14-6, it follows that 

Ei = C\-*-f(\T). (14-8) 

12. Distribution Laws. From considerations based on class- 
ical theory Wien derived a formula which gave an evaluation of 
/(XT), of the form 

= C,X-V-<'"* 7 ', (14-9) 



where Ci and C% are constants and e is the natural logarithmic 
base. It is evident that eq. 14-9 gives for a black-body radiator 
the distribution of energy as a function of wave-length. While it 
gives calculated values of E\ which agree with experimentally 
determined values quite well for wave-lengths in the visible 
region, the values for longer wave-lengths are too low. Slightly 



258 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIV 

different distribution formulas, obtained by various processes of 
deduction based on classical theory, were obtained by others, 
notably the Rayleigh formula 



(14-10) 

in which C\ and Cz are not necessarily the same constants as in 
Wien's formula. Rayleigh's theoretical treatment was later 
extended by Jeans, who proposed the formula 

J0x = Cm- 4 , (14-11) 

in which C is a constant and kT is the total energy associated 
with each degree of freedom of the medium in which the radiation 
is supposed to exist. 1 The formulas of Rayleigh and Rayleigh- 
Jeans agree with experiment for very long waves, but they give 
values which are too high in the visible spectrum. 

13. Planck's Quantum Hypothesis. Experimental tests car- 
ried out by Lummer and Pringsheim proved conclusively that 
none of the distribution laws completely agreed with observa- 
tions. Numerous attempts to modify the formulas derived from 
classical theory so as to bring complete agreement with experi- 
ment failed. Finally, in 1900, Planck decided to alter his method 
of deducing the distribution law by introducing a new and radical 
concept. In all previous deductions, the energy of radiation 
had been supposed to be divided among a great many hypo- 
thetical "oscillators" in the black body. In arriving at the 
mean energy it had been customary to suppose that an individual 
oscillator might possess any possible quantity of energy. 
Planck's radical departure consisted in postulating that the 
energy of the radiator was divided into a finite number of discrete 
units of energy of magnitude e, these energy units or "quanta" 
being distributed at random among the individual oscillators. 
The mftnber of different ways in which this distribution of energy 
may be divided among the oscillators was called by Planck the 
thermodynamic probability of a particular arrangement or dis- 
tribution. Making use of well-established rules of mathematical 

1 It should be kept in mind that no attempt is made here to give the 
student a complete understanding of the theoretical bases of these formulas. 
Fascinating as the development of the subject is, it is intended here only to 
introduce the formulas so as to contrast them with the more successful 
formula of Planck, whose work was part of the prelude to the quantum 
theory of spectra. 



SBC. 14-14] SPECTRA 259 

procedure, with the aid of this novel concept Planck arrived at an 
expression for the mean energy V of an oscillator: 

V - 



in which A; is a constant. But from classical considerations alone, 
Planck had deduced that 

E* = ]? (14-13) 

while it is evident from eq. 14-8 that the displacement laws, which 
are in rigorous agreement with experiment, hold that E\ is pro- 
portional to some function /(XT 7 ). Hence it follows that U is 
proportional to some function of (XT 7 ). Planck concluded that 
in view of the form of eq. 14-12, and the necessity for keeping in 
agreement with the displacement laws, it followed that 



= 



where v is the frequency of the radiation and h is a constant. 
The symbol h stands for Planck's quantum of action and has the 
value 6.547 X 10~ 27 erg sec. Planck's distribution formula is 



where Ci and c 2 * are constants. For short waves this is the same 
as Wien's formula, and for long waves is the same as the Ray- 
leigh-Jeans formula. 

14. The Rutherford Atom Model. During the time when the 
laws of radiation were engaging the attention of many research 
workers, and in the decade following, older concepts of the nature 
of matter were being subjected to rigorous scrutiny. The dis- 
covery of radioactivity and the development of the concept of a 
fundamental unit of electrical charge, the electron, stimulated 
experiments which showed conclusively that the older ideas 
concerning the structure of atoms must be modified. Finally, 
in 1911, on the basis of experimental results obtained in a lengthy 
investigation of the manner in which a-particles were scattered 

* c 2 is equal to hc/k, where c is the velocity of light, and k is Boltzmann's 
constant, the gas constant Ro divided by the Avogadro number No- k has 
the value 1.3708 X 10~ 16 erg deg.-. The value of c 2 is 1.432 cm.-deg. 



260 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIV 

by thin metallic foil, Rutherford postulated that an atom con- 
sisted of a positively charged nucleus of extremely small dimen- 
sions, surrounded by planetary negatively charged electrons, 
the distances between the electrons and the nucleus being very 
great compared to the dimensions of the charged particles 
themselves. 

16. The Bohr Theory of Spectra. At first it seemed as if the 
Rutherford atom model would lead the way on classical grounds 
to a solution of the riddle of emission (bright line) spectra, for it 
can be shown that the frequency of revolution of the planetary 
electrons depends upon the energy of the atom. On the other 
hand, the radiation of any part of the energy would necessarily 
lead to a gradual change of that frequency, and hence to the 
emission of a continuous spectrum, and not the separate line 
spectra which are observed. The problem, for the hydrogen 
atom at least, was finally solved by Bohr, who assumed the 
Rutherford atom model, and in addition made three hypotheses 
concerning the manner in which the radiation takes place. 

Bohr assumed, first, that the planetary electrons revolve about 
the nucleus, not in all possible paths but only in certain discrete 
orbits. He assumed that the orbits are circular and are 
limited to those for which the angular momentum is an integral 
multiple of h/2ir, where h is Planck's quantum constant. He 
assumed, second, that no radiation takes place while an electron 
remains in one of these orbits, but only when it passes from a 
given orbit to one of lesser energy, i.e., to one of smaller radius. 
He assumed, third, that when the electron passes from one of these 
quantum orbits of energy TF 2 to one of lesser energy W\, radiation 
is emitted whose frequency, v, i,s given by 1 

c Wi- 

v = x = -~h 



The calculation of the values of Wz and W\ for hydrogen is 
simple. If a is the radius of a given orbit, e the charge on the 
electron, and E the charge on the nucleus, then the force of 
attraction is eE/a?. For equilibrium this must be equal to the 

1 In making this assumption, Bohr was adopting not only Planck's 
hypothesis, but a far more drastic one proposed by Einstein to explain the 
photoelectric effect (see Sec. 15-20). 



SEC. 14-15] SPECTRA 261 

centrifugal force on the electron, which is mow 2 , where m is the 
mass of the electron and o> the angular velocity, or 

ma 3 o> 2 = eE. - (14-16) 

If Z is the atomic number, then E = eZ. Also, by Bohr's 
first hypothesis, the angular momentum is a multiple of h/2ir, 
or 

(14-17) 

where n is an integer. From eqs. 14-16 and 14-17 it is possible 
to obtain values of a n and , the radius and angular velocity 
corresponding to orbit n. 



and 



C0 n 



(14-18) 



The total energy W in an orbit is the sum of the kinetic and 
potential energies, or 



IT/ 2 2 

W = prWa 2 0> 2 -- ; 

2 a 

which by the use of eq. 14-16 becomes 

p*Z 

W = -~ 
2a 

Combining this expression with the first of eqs. 14-18, it follows 
that 



Combining this expression with eq. 14-15, we obtain finally 



' - A - ^rfe - i) (14 - 19) 

This may now be compared with the Rydberg formula, eq. 14-3. 
While v is the wave number, v is the frequency, so that vc = v. 



262 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIV 

Hence it follows from eqs. 14-3 and 14-19 that the Rydberg 
number is 

2ir 2 e 4 w 



a result which may be verified by calculation, given 

e = 4.770 X 10- 10 e.s.u. 
m = 9. X 10~ 28 gr. 
h = 6.547 X 10~ 27 erg sec. 
c = 2.998 X 10 10 cm. sec.- 1 . 

Although the Bohr formula gives values of the wave-lengths 
agreeing well with experiment for hydrogen, much more extensive 
hypotheses have been necessary to formulate a quantum theory 
Df spectra which holds for all atomic and molecular radiation. 
It is found, for instance, that the concept of circular orbits fails. 
The existence of discrete energy states, however, is completely 
verified, as is also the concept, expressed in eq. 14-15, that the 
Frequency of a given spectrum line is proportional to the differ- 
ences of energies of the beginning and end energy states involved 
in a transition of an electron from a higher to a lower energy 
state. 

It was pointed out in Sec. 14-4 that the Rydberg number is 
nearly the same for all elements. The slight variation is because 
of the effect of the mass of the nucleus, which has been neglected 
in the preceding discussion. From elementary mechanics it 
Follows that instead of m, the mass of the electron, the quantity 
mM/(m + M) t where M is the mass of the nucleus, should be 
used in eq. 14-19. Then the value of R conforms more closely 
to the experimental values. The value of M may be calculated 
from 

_ atomic wt. of element X mass of oxygen atom 

___ 

The quantity mM/(m -f- M) is known as the reduced mass. It 
approaches m as M approaches . 

16. Energy-level Diagrams. In the foregoing sections an 
explanation has been presented of the manner in which the simple 
Bohr formula in terms of physical quantities such as the charge 
on and mass of the electron, Planck's constant, and the atomic 
number, may be applied to give the wave-lengths of atomic 



SBC. 14-16] 



SPECTRA 



263 



spectral lines. These wave-lengths, or, more precisely, wave 
numbers, are expressed in terms of the differences between energy 
states of the atom, and the energy states, in turn, are identified 
with circular orbits in which satellite electrons move. In the 
preceding section it was indicated that certain concepts of this 
picture as, for instance, the concept of circular orbits, have been 
found to be not in agreement with observation. Nevertheless, 
the essence of Bohr's assumption remains that radiation of a 
particular frequency corresponds to a transition of the electron 
from a higher to a lower energy state. The energy associated 



n 











- 1 1 111 ' ^ 










I m r 
\ 

'' * " * T 










J 

!?!. 






c 
5 

*> 


* t 

*.< 

s. 

4*' 


. ! ! ! i 

1 I 1 I 1 

* t 



FIG. 14-4. Simplified energy-level diagram for hydrogen. Each arrow 
pointing downward indicates a possible transition in which radiant energy is 
given out by the atom. In a diagram to correct scale, the length of each vertical 
line corresponds to the frequency of a spectrum line. 

with each state is no longer considered necessarily as the motion 
of the electron in a circular orbit, but is energy of a certain special 
mode of motion of a satellite electron. It is possible to plot the 
energy states, and the transitions corresponding to spectral lines, 
graphically in what is called an energy-level diagram. 

Energy level diagrams representing the energy states and 
transitions for heavy atoms are sometimes complicated. That 
for the hydrogen atom is simpler, and is shown in Fig. 14-4. The 
horizontal lines represent on an arbitrary scale the different 
quantities of energy possible to the hydrogen atom. Radia- 
tion of a particular frequency v is represented by a vertical line 
drawn from an upper to a lower energy level. This is in agree- 
ment with Bohr's third postulate, which is that hv = W n - W m , 



264 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIV 

where W n and W m are the higher and lower energy states, 
respectively. 

17. Band Spectra of Molecules. In addition to the so-called 
line spectra due. to emission of radiation by atoms, there is another 
type of spectrum due to radiation by molecules. The former 
consist in the main of spectral lines which for any particular 
series are far apart, while the latter, called band spectra, appear 
with sufficient dispersion to be bands of closely grouped lines 
having a definite regularity in their spacing and, in many cases, 
with insufficient dispersion, to be continuous. A band is charac- 
terized by a head) either on the violet or red side, the lines there 
being usually so close as to be indistinguishable, and becoming 
gradually more widely spaced and fainter toward the tail (see 
Fig. 14-6). Bands having the head on the violet side are said 
to be degraded toward the red, while those having the head on the 
red side are degraded toward the violet. For heavy molecules 
such as Mn0 2 , 84, etc., the bands are more closely spaced than 
for light molecules such as CO, N2, etc., while the series of bands 
belonging to H2 are so widely spaced and overlap to such an 
extent that the system resembles an atomic spectrum. 

As in the simple Bohr theory of atomic spectra, the energy 
states in the molecule are limited to those for which the angular 
momentum due to its rotation is an integral multiple of h/2ir. 
Using j for the running number indicating different states 
(i.e.,j 0, 1, 2, etc.), and deriving from mechanics the expression 
for the angular momentum of the molecule, it may be shown 
that the energy W r due to the rotation in any state is given by 



W --, 

" r ~ o >>' 



h'f 

Sir 2 /' 



where / is the moment of inertia. The difference between values 
of W r for different values of j will represent a change in rotational 
energy. But changes may also take place in what might be 
called the internal energy, Wim of the molecule, owing to changes 
of the electronic orbits in the atoms, and to vibrations of the 
atoms within the molecule. For this reason, the total energy W 
is given as 



W = + Tftot - (14 " 20) 



SBC. 14-17] SPECTRA 265 

One of the problems in the analysis of band spectra is to estab- 
lish the relationship between the frequencies of the spectral linos 
in a band and the transitions between energy states in the 
molecule. According to quantum theory, the frequency of the 
line is given by v in 

hv = W - W", (14-21) 

whore W is the higher, and W" the lower energy state. Thus 
the frequency may be found by substituting values of W from 
eq. 14-20 into eq. 14-21. Before doing this, however, it should 
be stated (a) that modern quantum theory substitutes j(j -\- 1) 
for.; 2 , and (b) that the value of / may also change during a transi- 
tion, because of a change in the size of the molecule. With these 
details in mind, we obtain 



. _ W - W" _ h \j'(j' + 1) _ j"(j" + 1)1 
v he 8^[ /' I" J "*" 

(14-22) 



It should also be stated that changes in rotational and internal 
energy are independent. The change in the internal energy 
accounts for the location of the band in a particular region of the 
spectrum; the change in rotational energy, for the line in the 
band. The quantities / and /' are quantum integers which 
according to certain selection principles have a difference 
Aj(=/ j") which can only have the values 1, 0, or +1. 
A further condition holds that neither j' nor j" can be less than 
zero. 
Putting 



' W "1 = A 

int * ' iut J "> 




- = B 

Til *** 

lUTT-CI 1 

and 

h fl __ 
/" 



the three possible values of Aj give three series of wave numbers 
conventionally designated as 



266 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIV 



R(j) = A + 2B(j + 1) + C(j + 1)' j - 0, 1, 2, , 
QO) = A + Cy + Cy 2 j = 0, 1, 2, . . , 

Ptf) = A - 2y + Cy 2 j - 1, 2, 3, . 

These three equations therefore describe three branches, P, Q, 
and Rj which exist in any electronic band for which the conditions 
given above hold. Figure 14-5 is a diagram in which the values 
of wave numbers are shown as plotted as abscissas against the 
quantum numbers j as ordinates. In the case illustrated /' > 7", 




FIG. 14-5. At the bottom are shown the wave numbers and intensities of 
individual lines in the branches of a band system, while above they are plotted 
against quantum number j. The band represented is degraded toward the red. 

and the band degrades toward the red. This diagram shows that 
the head is simply the position of the turning point of one of the 
branches. The greater intensity of the head is usually because 
of the close grouping of the lines at that point, although some- 
times the lines in that part of the branch are also more intense. 

In Fig. 14-6 are shown several typical band spectra with dis- 
persion permitting illustration of the structure described above. 

18. Continuous Absorption and Emission by Atoms. Ordi- 
narily the absorption of white light by atoms of a given element 
results only in line absorption such as that found in solar and 
stellar spectra, but under proper conditions it is possible to 
produce continuous absorption also. If the frequency v of the 
light incident upon the absorbing gas is sufficiently great, the 



SBC. 14-18] 



SPECTRA 



267 



energy communicated to some of the atoms will be sufficient to 
eject an electron completely, causing ionization. The absorption 
will take place at a wave-length shorter than that corresponding 
to w 2 = in eq. 14-3. This is the same as saying that when the 




Fio. 14-6. Photographs of band spectra, (a) Some bands of the NO mole- 
cule (Mulliken) ; (6) the CN band at 3883 angstroms (Mark Fred) ; (c) bands of 
F: molecule (Gale and Monk) with low dispersion and an iron comparison 
spectrum; (d) one of the Ft bands with high dispersion. 

n in the first of eqs. 14-18 becomes infinite, the value of a n 
becomes infinite, and the Bohr theory ceases to apply, so that all 
energies are possible. Since all such high frequencies will thus 
eject the electron, there will be a continuous absorption band to 
the violet of the convergence frequency of the line series. 




FIG. 14-7. Line absorption and continuous absorption by the sodium atom. 
Decreasing wave-length toward the right. On the left of the arrow indicating 
the convergence limit are several members of the spectrum of sodium; on the 
right is the continuous absorption. (Photograph by G. R. Harrison.) 

It should be stated that, while the discussion strictly applies 
to the hydrogen atom, it applies to the general case with some 
changes of quantum details. Continuous absorption is illus- 
trated by the photograph of the sodium spectrum in Fig. 14-7. 



268 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XIV 



1 



In a similar manner, a continuous emission spectrum may be 
produced, by the radiation of energy by atoms to whose finite 
energy levels an ejected electron has returned from orbits other 
than those represented by a series of finite integers w 2 - 

The energy level diagram representing these cases is shown in 
Fig. 14-8, the shaded portion at the top representing the con- 
tinuum of energy levels greater than n 2 = <. A continuous 

absorption by atoms in a gas is 
represented by arrows from dis- 
crete levels and ending in the 
shaded area; and continuous 
emission by arrows starting in 
the shaded area and ending at 
the discrete levels. 

19. The Structure of Spectral 
Lines. General. A spectral 
line owes its characteristics 
mainly to three things: (a) It is 
an image of the source, which is 

E j T usually a very narrow slit ; (b) it 

Fi. 14-8. Energy level diagram depends upon the character of 

for emission and absorption. Vertical tne diffraction or interference 1 
lines E represent absorption by excita- 11,11- 
(ion; I, ionization; T, types of emis- pattern produced by the disper- 
sion; the line downward from the s j ve instrument; (c) it depends 

continuum to level 1, continuous . . 

emission. upon causes inherent in the 

source. It is the third of these 

which is to be discussed here. The conditions in the source 
may result in a broadening of the line, either symmetrical 
or asymmetrical, or it may split up th'e line into a complex of 
lines. 

20. The Broadening of Lines, a. The Natural Breadth of a 
Line. In quantum mechanics, the discrete energy levels postu- 
lated by the Bohr theory are considered rather as the locations 
of maxima in a probability distribution of energy changes in the 
atom. This may best be visualized by considering the horizontal 
lines representing levels in such a diagram as Fig. 14-4, not as 
infinitely thin lines but having width and a density distribution. 
A spectrum line due to like transitions in many atoms will, 
therefore, have a width and shape depending on the character- 
istics of the two levels involved in the transition. 



SEC. 14-21] 



SPECTRA 



269 



6. The Doppler Broadening. This is due to the random motions 
of the radiating atoms and hence is often subject to experimental 
control. It can be shown that the "half width" of a line, as 

shown by Fig. 14-9, is given by 1.67-, 



where v is the 

c\ u ' 

frequency for no motion of the atom, c is the velocity of light, 
R is the universal gas constant, T is the absolute temperature, 
and u the molecular weight. 

c. Breadth Due to Collision. It is assumed that while an atom 
which is absorbing or radiating energy collides with another 
atom, the phase and amplitude 
of the radiation may change. 
This leads to a half width equal 

IRT 



to 4Nr*d< 



where N is the 



Avogadro number, d the density 
of the gas, r the average distance 
between nuclear centers when 
closest, and R y T, and u have 
the same significance as before. 
d. Broadening and Asymmetry 
Due to Pressure. It is found that 
increasing the pressure 




FIG. 14-9. Illustrating the half-width 
of a spectrum line. 



on a 

radiating gas causes an unsym- 
metrical widening of the line and a shift of its maximum toward 
longer wave-lengths. This may also be considered as due to the 
interaction of the electric fields of the atoms and ions in a dis- 
charge, and thus, as a broadening due to the Stark effect (see 
Sec. 21d below). 

21. The Complex Structure of Lines, a. Fine Structure 
(MuUiplet Structure). An electron in a given energy state has 
orbital motion and spin motion. The angular momenta of these 
may be coupled in different ways, giving rise to a splitting of the 
energy levels postulated in the simple Bohr theory into sublevels. 
For instance, if a given level, n = 2, is divided into two, rather 
close together, there will be two spectral lines instead of one, 
as in the case of the sodium doublet 5890 and 5896. The spacing 
of these multiplets, i.e., doublets, triplets, etc., in the spectrum 
increases with atomic weight, being so small for the lighter 
elements that the lines appear single except with the highest 



270 



LIGHT: PRINCIPLES AND EXPERIMENTS (CHAP. XIV 



resolution. The fine structure separation for the two com- 
ponents of the red hydrogen line 6563 angstroms is only 0.14 
angstrom. On the other hand, the multiplets of heavier ele- 
ments may be separated by over 100 angstroms. 

6. Hyperfine Structure. As the name suggests, this Is a 
complex structure of much less separation than fine structure. 



r 





(a) 



1 1 

1 


: i ; : 




1 1 


200 


\ /? ,,.,,^ 


204 



202 
Isotopes 

FIG. 14-10. Hyperfine structure of the green mercury line, 5461 angstroms. 
Above, photograph with the second order of a concave grating, 30-ft. radius of 
curvature, and 30,000 lines per inch, ruled by Gale at Ryerson Laboratory. 
Below, the theoretical hyperfine spin and isotope structure. The displacements 
in wave number from left to right are -0.765, 0.468, -0.315, -0.093, -0.064, 
-0.037, 0, +0.020, +0.121, +0.195, +0.305, +0.753 cm.-i. Visually, all but 
the strong central five components are easily resolved with the grating used. 

Hyperfine structure is due to two causes. One of these is the 
presence in a source of more than one isotope of an element, giving 
rise to a line for each nuclear mass; the other is the spin of the 
atomic nucleus. Figure 14-10a is a photograph of the mercury 
green line, 5461 angstroms, while 6 shows the theoretical structure. 
In this case both causes of hyperfine structure are present. 

c. The Zeeman Effect. In a strong magnetic field, a single line 
is split up into many components whose separation depends on 



SEC. 14-21] SPECTRA 271 

the strength of the field. This effect is to be discussed in detail 
in Chap. XVI. 

d. The Stark Effect. An effect similar to that discovered by 
Zeeman is produced if the source is in a strong electric field. 
This effect will also be discussed in Chap. XVI. 

Problems 

1. Using the simple Bohr formula, calculate the wave numbers of 
the first five members of the spectrum emitted by ionized helium, i.e., 
helium atoms which have lost one satellite electron by ionization and 
are radiating with the second. Make a chart of the energy levels and 
transitions for these five lines. 

2. Taking R to be 109,677.7, and using eq. 14-2, calculate the values 
of v and X for the convergence of the Balmer series, i.e., the value of X 
for which n = <*>. From the relationship hv = eV, where e is the 
charge on the electron, calculate the value of V in volts required to 
excite this line. 

3. Considering the sun as a black-body radiator with a surface 
temperature of 6000 abs., compute the total energy in ergs radiated 
by it in one year. How much of this is intercepted by the earth? 

4. Calculate the radius of the normal orbit (n = 1) of an electron 
in a hydrogen atom. Calculate also its velocity along the orbit, and its 
total energy. 

6. The atomic weight of hydrogen is 1.0082, and of deuterium, 
(heavy hydrogen) 2.01445. Compute the separation in wave number 
and wave-length of the first two lines of the hydrogen and deuterium 
Balmer series. 

6. If the value of X max for a black body is 5000 angstroms, what is its 
absolute temperature? At this temperature, what energy in ergs does 
it radiate in an hour for each square centimeter of surface? 

7. The reduced mass may be represented by 

m 



where A is the atomic weight and Mh is the mass of the oxygen atom 
divided by 16. Calculate the values of R for hydrogen, deuterium, 
helium, oxygen, copper, and silver. Plot them against the atomic 
weights. To what value of R (called R*) is the curve through the 
plotted points asymptotic? 

8. When an electric-arc light between terminals of, say, iron is enclosed 
in a chamber and subjected to an atmospheric pressure of several 
atmospheres, many of the spectrum lines are widened and show self- 
reversal (by self-reversal is meant the appearance of a sharp dark center 
to the bright emission line). Explain these two effects (see Sec. 14-lc). 



CHAPTER XV 

LIGHT AND MATERIAL MEDIA 

In all the preceding chapters except Chap. XIV, which traced 
the rise of the quantum theory of spectra, the principles dealt 
with have been those which concern only the light itself, no 
account being taken of its interaction with material media through 
which it passes. This is true even in the case of the treatment of 
prismatic dispersion and chromatism, for there was no further 
discussion of the nature of the media, except that the refraction 
of the light took place in accordance with Snell's law. In the 
present and following chapters, the nature of the media will be 
taken into account. The subject is far too extensive for an 
exhaustive discussion, which would, indeed, be out of place in an 
intermediate text. It is the intention, however, to present it 
fully enough to give the student an introduction to modern 
theories of the interaction of light with media through which it is 
transmitted. 

1* Absorption. Light energy incident upon the surface of a 
medium undergoes absorption, refraction, reflection, or scatter- 
ing. A large part of the energy absorbed is changed to heat or 
chemical energy. Some substances absorb light of one wave- 
length group and afterward emit light of another, almost invari- 
ably greater, this phenomenon being known as fluorescence. 
For some substances the absorption is general, i.e., it is the same 
or nearly so for all wave-lengths. Others exhibit selective absorp- 
tion which is more or less complete for certain spectral regions 
while for others the transmission is very high. Among sub- 
stances showing general absorption are thin metallic films, lamp- 
black, and metallic blacks which are composed of finely divided 
particles of pure metal. These "blacks" will be discussed later. 

The absorption by a gas of energy corresponding to those 
frequencies which atoms may emit has been discussed in Chap. 
XIV. 

2. Laws of Absorption* If a beam of light of intensity 7 is 
incident upon absorbing material, it may be said that each ele- 

272 



SEC. 15-3] LIGHT AND MATERIAL MEDIA 273 

ment of thickness, or each layer, absorbs the same fraction of the 
light passing through it. Then the intensity / of the light after 
passing through a total thickness d is given by 

/ = he-*" 1 , 

where /i is the coefficient of absorption of the material and e is the 
Napierian logarithmic base. This is known as Lambert's law. 
The value of M depends upon the wave-length of the light. 

A similar relation has been proposed for absorption by solu- 
tions. Here the absorption depends not only upon the thickness 
d traversed but also upon the total number of absorbing mole- 
cules and hence upon the concentration, whereupon we have 



-Act! 
l 



in which A is the absorption coefficient for unit concentration, 
or the molecular-absorption coefficient, and c the concentration. 
This is known as Beer's law. While Lambert's law is upheld by 
observations, Beer's law does not always hold, since in some cases 
A varies with the concentration. 

3. Surface Color of Substances. Most substances owe their 
surface color to selective absorption. Internal reflections and 
refractions take place beneath the surface between the particles 
as well as absorption by them, the light which is finally returned 
from the substance being that which is least absorbed. There- 
fore an object which is red because of selective absorption of 
shorter wave-lengths will appear almost black if illuminated 
only with blue light. A small amount of color is, of course, 
reflected from the particles in the outermost layer. 

The internal reflections and refractions which thus account 
for color would not take place if the medium were homogeneous. 
It is evident therefore that a mixture of two colored pigments 
does not produce the same color that results when two light 
beams of the same two colors are combined. While in the latter 
case the eye receives the actual wave-lengths in the combination, 
in the former the absorption of the mixed pigments is not neces- 
sarily the sum of the separate absorptions. 

A substance which appears white has that property because 
it is composed of finely divided transparent particles which are 
either not in optical contact, as in the case of powdered glass or 
crystal, or are embedded in a transparent medium of different 



C 80 
o 
"o. 
o 



60 



e 



40 
8 



274 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XV 

index of refraction, as in the case of white paint. If either the 
finely divided powder or the surrounding medium have selective 
absorption, the paint will appear not white but colored. 

4. Color Transmission. The color transmitted by an absorb- 
ing medium depends mainly upon the selective absorption. If 
the medium is homogeneous, so that the same fraction is absorbed 
by each unit layer, the intensity of the transmitted light is given 
by /a', where 7 is the intensity of the light refracted into the first 
surface of the medium, t is the thickness, and a is the transmission 
coefficient. 

Certain aniline dyes exhibit a characteristic of transmission 
called dichroism (or dichromatism). For instance, while the 

light transmitted by a concen- 
trated solution of cyanine 
appears red, that by a dilute 
solution is blue. Also, with 
greater concentration there is 
an increase in the purity 1 of 

r / 

the color. This is so because 

a20h \ the coefficient of transmission 

for the blue is smaller than that 

4000 ""~~~50oo ^eooo TOGO ^ or * ne roc ^ while at the same 

Wavelength in Angstroms time the visual sensation due to 

Fio.16-1. T ] = transmitted light; D - tho W j s groat er than that 
diffused light. 

lor the red. Because of the 

latter factor, with small concentration blue will predominate, in 
spite of the small transmission coefficient, while great concentra- 
tion will practically absorb all the blue, leaving only red with 
increased purity. 

Similar results are obtained with thick and thin layers of sub- 
stances exhibiting dichroism, provided the concentrations are 
equal. 

Scattering is also partly responsible for color of transmission, 
especially in colloids, although even there it plays a minor role. 
In Fig. 15-1 are shown the relations, for a colloidal solution of gold, 
between the coefficient of absorption, the transmission, and the 
scattering of light. 

! Purity is the ratio of the luminosity of the dominant monochromatic 
radiation to the total luminosity. 



4J 




SEC. 15-6] LIGHT AND MATERIAL MEDIA 275 

5. Absorbing Blacks. In Sec. 16-1 was mentioned the prop- 
erty of general absorption possessed by certain metallic blaeks, 
such as lampblack, platinum black, chemically precipitated silver, 
and other metals in a finely divided state. These substances 
which when in solid blocks exhibit high reflection, either specular 
if polished or diffuse if rough, owe their peculiar blackness to the 
finely divided state. In this state the lampblack or metal is 
really a mass having great porosity. The light is reflected 
mostly into the open spaces between the particles, with a partial 
absorption at each reflection. The ideal arrangement of particles 
for this effect would be an array of needle-shaped highly absorbing 
particles all on end to the surface, as in the case of black velvet. 

6. Early Theories of Dispersion. Reference has already been 
made to Cauchy's dispersion formula, which is simply an empirical 
relation of the form 

n = n + , + 4 + (15-1) 

in which rio, B, C, etc., are constants depending on the substance. 
This formula tolls us nothing of the nature of that substance, 
nor of its interaction with the light passing through it. More- 
over, it is not in agreement with the facts in all cases. For sub- 
stances which are not transparent for all wave-lengths but show 
selective absorption, the index of refraction, n, does not increase 
continuously as the wave-length decreases, as required by Cau- 
chy's formula. Instead, for wave-lengths slightly shorter 
than those in the region of absorption, the index is less than for 
wave-lengths slightly greater than those of light absorbed. The 
effect, known as anomalous dispersion, can be examined by 
means of successive dispersion by two prisms whose refracting 
edges are at right angles.. 1 

A formula, due to Sellmeior representing the situation, is 



' 

The particles of the substance are supposed to possess a natural 
period of vibration whose frequency corresponds to X. D is a 

1 For detailed descriptions of such experiments the reader is referred to 
Wood, "Physical Optics," Editions of 1911 and 1934, Macmillan. 



276 



LIGHT: PRINCIPLES AND EXPERIMENTS (CHAP. XV 



constant. In the case of a substance which shows more than one 
absorption band, the formula may be written 



(15-3) 



A graphical description of the results for two absorption bands 
is given in Fig. 15-2. 

Sellmeier's formula is an improvement on that of Cauchy 
because it gives more accurate values for n as the region of an 
absorption band is approached. -In regions very close to the 

absorption band, however, it cannot 
be applied, since n becomes infinite. 
In other regions it represents the 
experimental results very well. 
Helmholtz proposed a mechanical 
formula based upon the supposition 
that the atoms, being capable of 
vibration about fixed positions with- 
in the molecule, were subject to 
vibrations due to the oscillatory 
motion of the light wave transmitted 
through the medium. In order to overcome the difficulty 
inherent in Sellmeier's formula, and to account for absorption, 
he supposed also that the vibration of the atom was attended by a 
damping force of a frictional character. The theory of Helm- 
holtz was extended by Ketteler, who produced a modified formula 
containing a term for the index of refraction for very long waves. 
Both the Helmholtz and the Helmholtz-Ketteler dispersion 
formulas are found to agree within limits with experimental 
results. Although the latter formula is not unlike that obtained 
on the basis of the electromagnetic theory, and involves a term 
for the dielectric constant of the medium, for purposes of com- 
parison with eq. 15-3 it will be given in the simplified form 



(15-4) 




where D f is a constant, and CrX 2 the term representing the fric- 
tional force. 1 



1 For a good summary of the earlier theories of dispersion see Preston, 
'The Theory of Light," Macmillan. 



SEC. 15-7] LIGHT AND MATERIAL MEDIA 277 

7. The Electromagnetic Theory of Dispersion. The formulas 
given in the preceding section were either empirical or based upon 
the assumption of material particles possessing natural vibration 
periods, and set into oscillation mechanically by the light wave 
While the generality of the Helmholtz-Ketteler formula was 
greater than that of any which preceded it, particularly because 
of the inclusion of the damping term, its fundamental assump- 
tions were not in harmony with the electromagnetic theory and 
the electron theory of matter. The basis of a more rigorous 
electromagnetic theory of dispersion was laid down by Drude and 
Voigt, 1 and later was brought into harmony with modern 
theory. 

In this theory a concept of a damping factor was introduced 
as in that of Helmholtz. According to the modern electron 
theory of matter, atoms consist of positively charged nuclei and 
negatively charged electrons. In the electric field of the light 
wave these are set into oscillation. The idea may be illustrated 
by considering a body carrying positive and negative charges to 
exist in the electric field of a condenser. Owing to the field, 
the negative charges will be displaced toward the positive plate 
of the condenser and the positive charges toward the negative 
side resulting in an induced dipole moment in the body. If the 
condenser is discharged, the dipole will be set into oscillation. 
A similar picture holds for the effect of the electromagnetic light 
wave upon the atoms of the substance through which the light 
passes. An equation of motion, which includes also a damping 
term to account for the absorption of light energy, may be set up. 
This equation leads to a solution which may be expressed in terms 
of the dielectric constant of the medium. The manner in which 
this may be related to the index of refraction is as follows. 

According to the electromagnetic theory, for frequencies of 
vibration as great as those of light, the index of refraction of a 
material medium is given by n \/e, where e is the dielectric 
constant, or specific inductive capacity, of the medium. If the 

1 See Drude, "Optics," Longmans, and Houstoun, "A Treatise on Light." 
For more modern presentations see Lorentz, "Problems of Modern Physics," 
Ginn; also Slater and Frank, "Introduction to Theoretical Physics," 
McGraw-Hill. A summary of theories of dispersion and a review of the 
quantum theory of dispersion is presented by Korff and Breit in Reviews of 
Modern Physics, 4, 471, 1932. 



278 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XV 

medium is absorbing, its index of refraction n is best given as a 
complex quantity, and may be written n = n(l IK) where K is 
the absorption index. l The resulting dispersion formula is 

e 2 

N- 



in which N is the number of electrons per unit volume, e the 
charge on the electron, ra its mass, v the frequency of the light, 
and v t the natural frequency of vibration of an electron. 

When v is not near v a , the frictional term G may be neglected, 
the right-hand side of the equation is real, and eq. 15-5 becomes 

i 

N 



Upon using the relation c = v\ to change from frequency to 
wave-length, eq. 15-6 takes the form 

ATe 2 X. 2 X 2 



which has the same form as Sellmeier's formula, eq. 15-3. 

On the other hand, for values of v very close to v, the frictional 
term is important and cannot be neglected. Considering the 
case for absorption by gases, for which n is very close to unity, 

1 Sometimes called the extinction coefficient, and sometimes the coefficient 
of absorption. The two are related, but not the same. In traversing per- 
pendicularly a thin layer of absorbing material of thickness d, the amplitude 



of vibration of light of wave-length X decreases in the ratio 1 : e , where 
K is the extinction coefficient. In consequence, the ratio of the intensities of 

-4 * 

the emerging and incident light is given by /i//o = e \ For an absorb- 
ing layer of thickness X, this ratio is given by /i//o = e"" 4 * 71 ", from which it 

follows that ic = T" log y. The coefficient of absorption, which may be 

called M, is related to the extinction coefficient by /u = 47nc/X, since / = he~* d . 
The term absorption index is preferred, because the word extinction implies 
complete dissipation of the light energy. 



SEC. 15-8] 



LIGHT AND MATERIAL MEDIA 



279 



and assuming only one natural frequency v,, eq. 15-5 may be 
written 



w 2 (l - w)* = 



2m 



Separating this into real and imaginary parts, we obtain 



2m 



and 



Ne' 2 



- / 2 ) 2 + GV 



6V 



(15-8) 



(15-9) 




2m (v, 2 - r 2 ) 2 + 

The values of n and K are plotted in Fig. 15-3. 

The results given above are for gases, in which each molecule 
is considered to be entirely free from 
the influence of others. In liquids 
and solids this influence must be 
taken into account. The result is a 
dispersion formula of the same form 
as eq. 15-5, except that the natural 
frequencies are different by a factor 
depending upon the effects of the 
molecules upon each other. 

8. The Quantum Theory of Dis- 
persion. From the preceding sec- 
tions on dispersion it is evident that 
on the basis of any classical model 
the index of refraction of a medium is given by a formula con- 
taining a term proportional to 

1 

P.* - V 2 ' 

where v is the frequency of the incident radiation and v s is the 
frequency of an oscillator whose character depends upon the 
particular assumptions involved. According to the quantum 
theory of spectra, however, this oscillation frequency is not 
that of the radiation, yet experiments show that there must be 
some intimate connection between the refraction, dispersion, 
and absorption of a medium. In terms of the quantum concept 
of the origin of spectra, there should, then, be some relation 
between the change of energy hv a ' in absorption (and hence the 



FIG. 15-3. 



280 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XV 

frequency corresponding to an absorption region of the spectrum) 
and the dispersion of the light. Experimentally it is shown 
conclusively that there are absorption bands in spectral regions 
of so-called "anomalous" dispersion. 

This discrepancy between the meaning of v, in dispersion 
formulas and the significance of v a for absorption in spectral 
theory was one of the indications that the quantum theory in its 
earlier form was not sufficiently comprehensive to account for 
a wide range of associated phenomena. Owing to the eiforts 
of many investigators in the past decade, there has grown out 
of this difficulty a more satisfactory theory known as quantum 
mechanics. So far as spectra are concerned, the older concepts 
of energy states and much of the complex mechanism of the 
older quantum theory are retained. With regard to dispersion, 
the concept outlined in Sec. 15-7, that the index of refraction 
depends upon the electric dipole moment acquired by the 
particles under the influence of radiation, holds as in the electro- 
magnetic theory. The quantum mechanics dispersion formula 
is the same in form as eq. 15-6, except that (a) the term v s 
no longer relates to the natural frequencies of vibration of 
the particles, but to frequencies associated with the transitions 
between energy states, and (6) the numerator comprises terms 
which depend upon the probabilities of the transitions. In 
addition, the more general quantum theory of dispersion 1 
accounts also for the existence in scattered radiation of the 
Raman effect, which is to be discussed in Sec. 16-19. 

9. Residual Rays. In 1896 E. F. Nichols, working in Rubens' 
laboratory, discovered that in the regions of wave-length 8.5 
microns (= 85,000 angstroms) and 20 microns, crystal quartz 
possesses metallic reflecting power; i.e., for those wave-lengths 
it is as good a reflector as is a polished metal surface for visible 
light. Nichols' work was quickly followed by investigations of 
other crystalline solids. The discovery of this property of 
selective reflection was of great importance, for in these same wave- 
length regions crystal quartz has pronounced absorption bands. 
Consequently the absorptive characteristics of solid transparent 

1 Originally developed by Kramers and Heisenberg, Zeitschrift far Physik, 
31, 681, 1925, and later derived from the general considerations of quantum 
mechanics. The mathematical theory involved is beyond the scope of this 
text, but may be found in any comprehensive treatise on quantum mechanics. 



SBC. 15-9] 



LIGHT AND MATERIAL MEDIA 



281 



substances may be determined by finding its residual rays 
(reststrahlen). Also, substances with this characteristic, used 
as reflectors, serve to isolate rather narrow bands of wave-length 
in the infrared, thus taking the place of niters for isolating such 
regions. 

Of considerable importance theoretically is the fact that 
observations on residual rays permit determinations of the charac- 
teristic frequencies of the absorbing substances, since these fre- 
quencies are evidently associated with the mechanism of 
absorption. 

In spectral regions at which ordinary transmitting substances 
exhibit high selective absorption the value of the absorption 
index K (eq. 15-5) may be sufficiently large compared to (n 1) 
so that the reflectivity 1 is considerably higher than for other 
wave-lengths. This correspondence between selective reflection 
and selective absorption of transparent substances has been verified 
by numerous experiments on residual rays. The table shows the 
wave-lengths of residual rays of maximum intensity and absorp- 
tion maxima for a number of solid substances. 



Residual rays of substances 


Residual 
rays, 
microns 


Absorption 
maximum, 
microns 


Lithium fluoride, LiF 


17.0 


32 6 


Sodium fluoride, NaF 


35.8 


40 6 


Sodium chloride, NaCl (rock salt). 
Potassium chloride, KC1 (sylvite) . 
Rubidium chloride, RbCl 


52.0 
63.4 
73.8 


61.1 
70.7 
84 8 


Potassium bromide, KBr 


81.5 


88.3 


Potassium iodide, KI 


94.1 


102 


Silver chloride, AgCl 


81.5 




Silver bromide, AgBr 


112.7 




Thallium chloride, T1C1 


91.9 


117.0 


Thallium bromide, TIBr 


117.0 




Thallium iodide, Til 


151.8 




Zincblende, ZnS 


30.9 




Fluorspar, CaF<> 


2.4, 31.6 




Quartz, SiOz 


8.5, 20.0 




Calcite, CaCO 3 


6.76, 28.0, 90.0 











1 This quantity is defined in the next section. 



282 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XV 

It is evident that the region of maximum absorption does not 
coincide exactly with the region of strongest intensity of the 
residual rays, the former being displaced toward longer 
wave-lengths. l 

10. Metallic Reflection. In the preceding paragraphs dealing 
with the characteristics of ordinary transparent and semi- 
transparent substances, it has been stated that there is apparent 
a relation between ordinary selective absorption arid the posses- 
sion by the substance of characteristic electronic frequencies of 
vibration. In the discussion of residual rays, it appears, further, 
that so-called transparent media often have the property of 
metallic reflection for certain wave-lengths in the infrared, and 
at the same time have strong absorption for those wave-lengths. 
Turning to a consideration of ordinary metallic substances, it is 
found, conversely, that for certain wave-length regions these 
may also act like transparent media. 

Transparent substances and metals are also at opposite 
extremes with regard to electrical conductivity. Most trans- 
parent substances are good dielectrics, i.e., they are poor con- 
ductors. The property of electrical conductivity has been 
found to be associated with the presence of so-called free elec- 
trons, which are not bound in fixed relation to the molecules or 
atoms as are the electrons, mentioned above, responsible for 
absorption bands, but which may migrate more or less freely 
through the metal in response to an electromotive force. The 
peculiar optical properties of metals, namely, their reflectivity, 
absorption, and transmission, are therefore dependent not only 
upon the bound electrons, but also upon these free electrons. 
For certain wave-length regions, therefore, a knowledge of the 
optical constants of metals may be obtained from a knowledge of 
the electrical conductivity. Theoretically also, it is possible to 
study the manner in which these free electrons act under the 
influence of electromagnetic waves of light. 

The reflectivity R of a metal is defined as the ratio for normal 
incidence of the intensity of the reflected to that of the incident 
light. This may be obtained for metals from Fresnel's equations. 
In eq. 13-12 the amplitude of the reflected light for normal 

1 For a discussion of the theory of residual rays the reader is referred to 
Max Born, "Optik." 



SEC. 15-10] LIGHT AND MATERIAL MEDIA 283 

incidence for vibrations perpendicular to the plane of incidence 
is given as 

n - 1 

a\ = a : T > 
n -f 1 

and in eq. 13-13, for vibrations in the plane of incidence, as 

6, = fc!LZ 
n + 1 

These equations are for transparent media. For metals, which 
absorb strongly, n must be replaced by n(l IK), as indicated 
in Sec. 16-7. Making the substitution in eq. 13-12 we obtain 

a\ _ 1 n -f- inn 
a 1 + n inn 

which, multiplied by its conjugate, gives the reflectivity 

(n - I) 2 + nV ( , 

' ( } 



and for transparent media becomes simply 



(n + ) 2 
From the electromagnetic theory of light it can be shown that 

n 2 K = *, (15-11) 



v 



where a is the electrical conductivity and v is the frequency of 
the light. From eqs. 15-10 and 15-11 and making use of assump- 
tions based on experimental results, it is possible 1 to obtain R in 
terms of a. Equation 15-10 may be put in the form 

R = l + 



/ j 

(n 4- I) 2 + nhi 2 

Also, we may make the assumption that for metals the absorption 
is very nearly unity. Putting K = 1, there results 



For very long wave-lengths it is found that n for metals is very 
much greater than unity, so that we may ignore all terms in the 



284 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XV 

denominator of eq. 15-13 smaller than 2n 2 . Also, in eq. 15-11, 
putting K = 1, we have n 2 = <r/v. Making these approximations 
and substitutions in eq. 15-13, we have 

R = 1 - -?==. (15-14) 



This simplified and only approximate relationship between the 
reflectivity and the conductivity of a metal does not apply 
below 5 microns. For copper, using infrared radiation of wave- 
length 12 microns, Hagen and Rubens found experimentally 
the value 1 R = 1.6 X 10~ 2 , while from the conductivity the 
calculated value is 1.4 X 10~ 2 . 

11. Optical Constants of Metals. It has been shown that the 
value of the index of refraction n and the absorption index K may 
be found in terms of the electrical conductivity of a metal. 
These quantities may also be found by direct optical experiment. 
It may be shown by the electromagnetic theory that incident 
plane-polarized light becomes, on reflection from the surface of a 
metal, elliptically polarized. The extent of this polarization 
depends upon the azimuth of the plane of vibration of the 
beam and its angle of incidence. It may be shown that the 
following equations 1 hold with a fair degree of precision: 



K = sin A tan 



cos 2\(/ 



. v.v.^} Arfyr 

n = sm <p tan ^ ; r ^-7; 

1 + cos A sin 2^ 



2/11 2\ -2 * 1 cos A sin 2\l/ 

w 2 (l + 2 ) = sin 2 <f> tan 2 ^ ~ - ~ 

1 + cos A sm 



(15-15) 



where A is the difference of phase introduced by reflection 
between the component of the vibration parallel, and that 
perpendicular, to the plane of incidence, and <f> is the angle of 
incidence. The angle ^ is called the angle of the restored plane 
of polarization 2 measured from the plane of incidence. Thus, 
when incident plane-polarized light is changed by reflection to 
elliptically polarized light, it may be changed to plane-polarized 
light once more by a X/4-plate or Babinet compensator, and 

1 The derivation of these equations may be found in Drude's "Theory 
of Optics." 

2 It will he recalled that the plane of polarization is perpendicular to the 
plane of vibration. 



SBC. 15-12] LIGHT AND MATERIAL MEDIA 285 

tan ^ is given by the ratio of the component (of the reflected light) 
parallel, to that perpendicular, to the plane of incidence. 
Methods of determining v and ^ are described in Experiment 21, 
for the case where A = ir/2. 

A value of the reflectivity R is found by substituting n arid K 
obtained from eqs. 15-15 in eq. 15-10. l 

12. The Scattering of Light by Gases. If a strong beam of 
white light is passed through a cloud of small particles of dust or 
condensed water vapor, the cloud takes on a color which depends 
upon the size of the particles. With the smallest particles the 
color will be blue, while with increasing size the light scattered 
will contain longer and longer wave-lengths until finally it is 
gray, or even white. At the same time, the light of the direct 
beam transmitted through the cloud will appear more and more 
red, until it cannot be seen at all. The same general effect may 
be observed with particles in suspension in a liquid. A simple 
experiment may be performed by mixing a weak solution of 
hyposulphite of soda (hypo) with a little dilute acid, causing a 
precipitation of sulphur. The aggregations of sulphur particles 
increase in size as the chemical action proceeds. Although the 
best method for demonstrating the effect of size on scattering 
is to project a beam of light from a strong source through the 
liquid, a simpler way is instructive. The mixture may be made 
in a large beaker or battery jar, and a 25- or 40-watt lamp 
plunged beneath its surface, taking care, of course, not to bring 
about a short circuit in the socket. After a minute or so, the 
image of the lamp takes on an orange hue which becomes more 
pronounced until it can no longer be seen through the side of the 
jar. At the same time the scattered light seen by looking at the 
side of the jar changes from a blue white to a yellowish white. 

The selective scattering of light by particles can also be seen 
in the smoke from a freshly lighted cigar, which is blue from 
the tip while that drawn through the cigar and exhaled, being 
made up of coagulations of carbon particles, is gray. The colors 
of sunsets in a cloudy sky are also due to the scattering of light 
by water drops and sometimes dust particles. Often the most 
lurid sunset reds may be seen in the neighborhood of a smoky 
industrial district. 

1 A good summary of formulas, data, and bibliography is given by 
J. Valasek in the International Critical Tables, Vol. V, p. 248. 



286 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XV 

Because of these common observations it was originally 
supposed that the blue color of a clear sky was due to minute 
dust particles in suspension in the upper atmosphere. It was 
shown by Lord Rayleigh that this is not the case, and that the 
sky owes its blue color to the scattering of light by the molecules 
of the atmosphere. An overcast sky is, then, gray or dull white 
because the light is scattered by water drops of larger size. Also, 
if there were no atmosphere, the sky would be absolutely black 
at all times except for those points where celestial objects would 
appear. 

That ordinary skylight contains very little red is shown by 
landscape photographs taken through yellow or red color filters 
with plates specially sensitized to the red. With even a pale- 
yellow filter a clear sky appears dark in a photographic print, 
paling to a lighter shade at the horizon. For aerial surveys of 
landscapes photographic plates specially sensitized to the infrared 
are used, since details ordinarily obscured by scattered light of 
shorter wave-lengths are thus brought out distinctly. In this 
manner, landscapes many miles distant have been photographed 
from aeroplanes. 

The scattering of light by small particles was studied experi- 
mentally by Tyndall. He showed that the larger the scattering 
particles, the larger proportion of longer wave-lengths the 
scattered light contained, i.e., the less blue it became. His 
experiments led him to the conviction that gas particles were not 
responsible for any of the scattering. The principles on which 
the scattering may be explained were first stated by Rayleigh. 
He showed that the sky owes its blue color to scattering of light 
by the molecules of the atmosphere, the intensity of scattering 
being proportional to the inverse fourth power of the wave-length. 
Rayleigh's published papers on this topic appeared through a 
period of almost half a century, and treat the problem in all 
details. His conclusions may be summarized briefly as follows: 1 

The molecules of a gas traversed by the incident light may be 
considered as sources of secondary waves. Each molecule acts 
on the light individually, i.e., as if unaffected by the presence of 
other molecules. Between the primary wave incident upon a 
molecule and the secondary wave given off from it there exists a 
definite phase relation. Because the molecules are distributed 

1 See Schuster, "Theory of Optics," 2d ed., p. 325. 



SEC. 15-13] LIGHT AND MATERIAL MEDIA 287 

at random, the phases of the individual scattered waves have no 
fixed relation to each other, except in the direction of propagation, 
where they will have the same phase. Hence, in order to express 
the intensity of the scattered light, the sum of the intensities of 
the individual scattered waves is taken instead of the sum of the 
amplitudes. The effect of all the molecules in a layer is arrived 
at by summing up the effects of Fresnel zones into which the layer 
is divided. The resultant vibration thus obtained is combined 
with the vibration of the incident wave, the result being a change 
of phase which may be considered as due to a change in velocity 
like that which occurs when light enters a refracting medium. 
This accounts for the entry of the index of refraction into the final 
formula. The expression thus obtained for the intensity of the 
scattered light is 

(1 + cos* 0, (15-16) 



in which* A 2 is the intensity of the incident light, n the index of 
refraction of the scattering gas, N the number of molecules per 
unit volume, and ft the angle at the molecule between the direc- 
tion of observation and the direction of propagation of the inci- 
dent light. Equation 15-16 holds only if the incident light is 
unpolarized. It appears that the intensity of the scattered light 
is inversely proportional to the fourth power of the wave-length, 
a relation which holds for liquids and solids as well as for gases. 
13. Polarization of Scattered Light. While Rayleigh's law 
for the intensity of scattering given in eq. 15-16 is essentially 
correct, it was shown by Cabanncs 1 that it is necessary to take 
into account a factor depending upon the state of polarization of 
the light. Experiment shows that if the incident light is unpolar- 
ized, the light scattered at right angles to the direction of propa- 
gation of the incident light is almost entirely plane-polarized, 
with the plane of vibration perpendicular to the common plane 
of the incident and scattered beams. This may be explained in 
the following way: Consider unpolarized radiation proceeding 
from source S to molecule ra (Fig. 15-4). We choose a direction 

1 A comprehensive discussion of the scattering of light is given by 
Cabannes: "La diffusion moleculaire de la lumiere." A very readable 
survey of the subject is contained in a small volume by Raman, "The 
Molecular Diffraction of Light," published by Calcutta University, 1922. 



288 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XV 

S' perpendicular to Sm in which the scattered light is to be 
observed. In accordance with the usual treatment of problems 
in polarization, the unpolarized beam is considered to be resolved 
into two components of vibration, one perpendicular to the plane 
SmS', the other in that plane. The direction of vibration of the 
second of these components is the same as the direction of propa- 
gation mS' of the scattered beam under observation and thus 
will contribute nothing to the light at S'. The light at S' should 
therefore be completely plane polarized with its direction of 
vibration perpendicular to the plane SmS'. The argument holds 
for any point of observation on a plane, containing mS r , to which 
Sm is normal. At points of observation as S" not in this plane, 




S" S' 

Fi<5. 15-4. 

the light should be partially polarized. Actual experiments 
show that the light scattered in directions perpendicular to the 
direction of propagation of the incident light is not completely 
polarized, for reasons which will be discussed later. The use of 
a double image prism such as a Wollaston reveals a strong com- 
ponent of vibration perpendicular to the plane of S and S' and a 
weak component parallel to it. 1 Cabannes finds that the inten- 
sity of scattering is represented more closely if the right-hand 
side of eq. 15-16 is multiplied by a factor 



6 - lp 

where p is the ratio of the weak (parallel) to the strong (per- 
pendicular) component of polarization. 

The existence of some unpolarized scattered radiation in a 
direction at right-angles to the direction of propagation of the 
incident beam is believed to be because some of the molecules are 

1 It should not be assumed that this means the presence of two plane- 
polarized beams, one perpendicular and one parallel to the mutual plane of 
propagation, but rather that the scattered light is a mixture of plane- 
polarized and ordinary light. 



SBC. 15-14] LIGHT AND MATERIAL MEDIA 289 

anisotropic. This term may be explained in the following manner. 
Suppose a molecule to consist of three atoms, one with a positive 
charge and two with negative charges, as in the case of carbon 
dioxide. As long as the geometrical center of the double negative 
charge coincides with the position of the positive, the molecule 
has no electric moment, but if this coincidence does not exist, the 
molecule is said to have an electric dipole moment. Also, if 
the centers of electrical charge do coincide, the imposition of an 
external electric field will cause a relative displacement of the 
charges, resulting in an induced dipole. We may consider the 
vibration of these induced dipoles to be the origin of the scattered 
radiation. Since the molecules are oriented at random, the vibra- 
tions of many of them will be at angles with the direction of 
vibration of the light incident upon them. Such molecules are 
said to be optically anisotropic, and their contribution to the 
scattering is responsible for that part of the light which is 
uupolarized. 

Accurate measurements of the intensity and state of polariza- 
tion of the light scattered by gases are extremely difficult. Not 
only is its intensity a minute fraction of the incident light, but 
it is often completely masked by the greater scattering from dust 
particles. It is also difficult to get rid of multiple reflections in 
the apparatus. l In much of the earlier work it is probable that 
improper collimation of the incident light gave spurious results. 
In accounting for the phenomenon of scattering in the atmosphere 
still other disturbing factors enter, such as the presence of a 
certain amount of light scattered by the earth's surface, and 
secondary scattering by the atmosphere. At the same time, 
scattering is of considerable importance, since in some details 
it depends upon molecular structure, and thus offers a means 
of investigating that structure. Also, as is evident from eq. 
15-16, it provides a method of determining N, the number of 
molecules per unit volume, and, from it, calculating the Avo- 
gadro number. 

14. Fluorescence. While irradiated with light, many sub- 
stances emit in all directions some of the energy of radiation 
which they absorb, the color of the light emitted by these sub- 

1 See an article by R. J. Strutt (Rayleigh the Younger), Proceedings of 
the Royal Soc. (London), 95, 155, 1918. 



290 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XV 

stances, which are said to exhibit the property of fluorescence, 
depending upon the substance and not upon the wave-length of 
the incident light. Radiation of short wave-length, such as 
ultraviolet light or x-rays, is particularly effective in producing 
fluorescence. The term owes its origin to the fact that the effect 
was first noticed in fluorspar, which emits a blue light when 
irradiated with sunlight. 

Among other common substances which fluoresce with a blue 
light are: paraffine wax, kerosene, benzene, some lubricating oils, 
an aqueous solution of aesculin, and an aqueous solution of 
quinine sulphate with a few drops of sulphuric acid added. A 
solution of chlorophyll in alcohol shows red fluorescence. Fluores- 
cene in solution shows yellow green, as does also uranium glass. 
When irradiated with x-rays or cathode rays, most glasses 
fluoresce, the color depending on the kind of glass. Ultraviolet 
light causes the cornea and lens of the eye and the teeth to 
fluoresce strongly, and, in smaller amount, the hair and nails also, 
the strength of the effect appearing to depend on personal char- 
acteristics, such as pigmentation. It has been observed that after 
passing through a solution which fluoresces, the light exhibits 
reduced power of exciting the same fluorescence, because of 
absorption of the exciting light. Thus a weak light falling upon a 
solution excites marked fluorescence only in the layer it first 
strikes. 

The fluorescent light is not of a single wave-length but a band 
with a pronounced maximum of intensity. It was formerly 
believed that the wave-lengths emitted were always longer than 
those of the radiation effective in producing the fluorescence, a 
conclusion reached by Stokes and known as Stokes' law. More 
recent investigations have shown that while Stokes' law is 
generally obeyed, the wave-length of maximum intensity of 
fluorescence is independent of the wave-length of the exciting 
light. The intensity of the fluorescence of any solution also 
depends upon the character of the solvent. 

15. Polarization of Fluorescence. It has been found that 
fluorescence of solutions is polarized. The degree of polarization 
in some cases depends upon the concentration and the tempera- 
ture. In general, the more viscous the solvent, the more strongly 
is the fluorescence polarized, probably because of the tendency of 
the solvent to hold the molecules in a fixed orientation. 



SEC. 15-17] LIGHT AND MATERIAL MEDIA 291 

In the case of isotropic substances, the polarization also 
depends upon the obliquity of emission. Some fluorescent 
crystals also exhibit peculiarities of polarization. No such 
degree of polarization exists in fluorescence, however, as in 
scattering of light, where almost complete plane polarization 
exists, the vibrations being at right angles to the incident beam. 

16. Phosphorescence. The term fluorescence is used when 
the process of emission goes on while the substance is being 
irradiated. Substances which continue to emit light for some 
time after the exciting light is removed are termed phosphorescent. 
The emission continues for different periods of time, depending on 
the substance and sometimes on temperature changes. Calcium 
sulphide continues its phosphorescence for many hours after the 
exciting radiation is removed, and is for this reason used as an 
ingredient in phosphorescent paint. 

Phosphorescence and fluorescence are difficult to distinguish, 
since the former persists in some cases only for an extremely small 
fraction of a second after the exciting light is removed. Actually, 
all solid fluorescent substances are phosphorescent. It is cus- 
tomary to limit the use of the term phosphorescence to the prop- 
erty exhibited by certain crystalline substances which contain 
impurities in the form of metallic particles. It is these particles 
which are responsible for the phosphorescence. In all other 
cases of so-called phosphorescence a better term is delayed, or 
persistent, fluorescence. 

Little is known of what is actually going on in a solid which 
absorbs light and fluoresces. It is believed that a photochemical 
process takes place owing to the absorption of light energy, the 
process later reversing with the accompaniment of light emission. 

17. Fluorescence in Gases. Rayleigh the younger has 
observed that the D-lines of sodium (5890 and 5896 angstroms) 
are emitted from a glass container of sodium vapor when it is 
irradiated by the light of the zinc line at 3303 angstroms. This 
is a case of true fluorescence, and is explained by the quantum 
theory of spectra in the following manner. 

The D-lines of sodium constitute the first member of the 
principal series, of which the second member is the doublet 
3302.3 and 3302.9. Upon being irradiated by light of that 
wave-length (of the zinc spectrum in the case quoted) the atoms 
of the sodium vapor absorb energy of radiation, thereupon under- 



292 * LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XV 



^*r 

I 



I 1 

e- 1 

li 



Fni. 15-5. 



going a corresponding change of energy. According to spectral 
theory, this change of energy consists of an electron passing from 
the lowest, or ground, level to an upper level, as shown in Fig. 
15-5 by the arrow pointing upward. The atom then passes 
to the lowest level of energy in two steps, the first corresponding 

to the emission of a red line, the second to 
the emission of the ZMines. While the D- 
lines were observed, the red line was not. 
It has been shown that the difference of 
_ energy corresponding to the missing red line 
is transformed into energy of motion, i.e., 
heat energy, by collision between atoms. 1 
Many other cases of fluorescence of atoms 
have been observed. While the fluorescent 
spectrum of liquids and solids is a continuous 
"~ band of some width, that of a monatomic 
gas or vapor is composed of lines. The 
character of the fluorescence of atoms varies greatly with the 
presence of inert gases, because of the energy changes due to 
atomic collisions with the molecules of the inert gas. 2 

Under certain conditions, an increase of density of a gas causes 
a decrease in the intensity of the fluorescent light. The explana- 
tion is that at the higher density the molecules or atoms have 
more opportunities for collisions with each other. The result is 
an increased proportion of the energy of the incident beam being 
changed into heat energy and a smaller amount being scattered 
as fluorescent light. 

18. Resonance Radiation. In the course of some experiments 
on the fluorescence of sodium vapor with white light, R. W. Wood 
limited the wave-length of the exciting light by means of a 
monochromator to a very narrow band at the region of the 
sodium D-lines. He found that the spectrum of the fluorescent 
light thus excited consisted of a number of single lines distributed 

1 Collisions of this sort, in which the potential energy possessed by excited 
atoms or molecules is given up to other atoms and thus changed to kinetic 
energy of agitation, are called collisions of the second kind. If a collision 
occurs between atoms or molecules by which one of them is raised to an 
excited state (i.e., an electron moved to a higher energy level), it is called a 
collision of the first kind. 

2 For an extensive discussion of fluorescence the reader is referred to 
Wood, "Physical Optics," 2d ed., Chaps. XVIII, XIX, XX, Macmillan. 



SEC. 15-19] 



LIGHT AND MATERIAL MEDIA 



293 



throughout the spectrum. The wave-length distribution 
changed with slight alterations in the exact wave-length range 
allowed to pass from the incident light by the monochromator. 
The fluorescence obtained in this manner he called resonance, 
radiation, and the spectra, resonance spectra. Resonance radia- 
tion may be obtained by the use of an irradiating source consisting 
of a single line of a metallic spectrum, and also with other vapors 
than that of sodium. ^ < 

19. Raman Effect.-(ln. 1928, Raman, after several years of 
investigation of light scattering, discovered that when a trans- 




Fio. 15-6. Raman effect in carbon tetrachloride. Above, the spectrum of the 
incident light. Below, the spectrum of the scattered light, showing the Raman 
lines on either side of the stronger lines due to ordinary scattering. (From 
Raman and Kri&hnan, Proceedings of the Royal Society of London, 122, 23, 1929.) 

parent liquid is irradiated with monochromatic light from a 
strong source the spectrum of the scattered light contains, in 
addition to the exciting line of frequency v, several weaker 
lines on either side, whose frequencies are given by v + Av. 
Later, the same effect was discovered in solids and gases. The 
differences AJ> are independent of the frequency of the original 
radiation and depend only on the nature of the scattering 
medium. The appearance of the displaced lines, known as the 
Raman effectfas illustrated in Pig. 15-6. As is evident, it is not 
necessary td use strictly monochromatic light provided the 
spectrum of the source contains only relatively few lines. 

The lines displaced to thejed are of tenjsferred to as StokesJLnes 
and those to the violet as anti-Stokes lineg.) This custom arose 
from the hypothesis proposed by. Stokes many years earlier, and 
referred to in Sec. 15-14, that secondary radiation such as 



294 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XV 

fluorescence was always of longer wave-length than the incident 
light. In the Raman effect the anti-Stokes lines are invariably 
fainter than the Stokes lines. 

The displaced lines are so much fainter than the lines of the 
exciting radiation that very long exposures are necessary to 
photograph them. ^A simple type of apparatus is shown in Fig. 
15-7. The source M is usually a quartz mercury arc of great 
intensity. The liquid to be examined is contained in a horn- 
shaped tube R, shielded from extraneous light and surrounded 
by a water cooler W, the small end of the horn being blackened 
and curved so that light reflected internally will be directed 
away from the larger end at which the observations are made. 




M 



Fio. 15-7.- A form of Raman tube. j 

The discovery of Raman was not entirely unexpected. In 
1923, A. H. Compton, while examining the spectra of x-rays 
scattered by a solid, discovered in the spectrum of the scattered 
radiation a line of smaller freqyency than that ofc^the incident 
x-rays. Also, in the same yearjdt had been predicted by Smekal 1 
that in addition to light of the same frequency as the incident 
radiation thorc should be present in the spectrum of ordinary 
scattered radiation lines with combination frequencies v v m , 
where v m is a characteristic frequency of absorption of the 
molecule, to be observed in the absorption spectrum in the infra- 
red. Smekal's suggestion was that when a photon of energy 
hv is incident on a molecule there will take place an exchange of 
energy in which the photon will either be augmented by, or 
have subtracted from it, an amount of energy hv m . In 1924, a 
similar prediction was made by Kramers 2 upon the basis of a 
new quantum theory of dispersion (Sec. 15-8), which was pub- 

1 Naturwssenschaft, 11, 873, 1923. 

2 Nature, 113, 673, 1924. 



SEC. 15-19] 



LIGHT AND MATERIAL MEDIA 



295 



lished in more complete form by Kramers and Heisenberg the 
following yearTI 

Raman's cEscovery seemed at first to be a complete con- 
firmation of SmekaPs prediction. Further observations soon 
disclosed that, although for many Raman lines the frequency 
differences Av (of the first paragraph of this section) agreed 
approximately with the frequencies in infra-red absorption 
bands, actually! Raman lines are often observed for which there 

AMMft ' X 4 

exist no corresponding observed absorption frequenciep. More- 
over, (jsome substances having strong absorption bands show no 
Raman lines with corresponding values of AJvt It was further 



1 
- 




\ 1 


(fit 


~_ 


- 


/ 


Scattering of hv 


Raman line h (p-4 V 9 ) 


Raman lineh(v+ A V f ) 




1 


- 





\ 


t 


\ /Vr I 


^ 


11 (/ 
f " fr} 



(g) 



FIG. 15-8. 



discovered that, evenj^in those cases where a rough agreement 
existed between the values of AJ> and the frequencies v m of absorp- 
tion bands, there was no agreement between the intensitiesTA 

Classical theories offer no satisfactory explanation of tnese 
observations. Those theories would require that the molecule 
of the irradiated substance have natural vibration frequencies v m 
which, combined with the frequency of the incident light, give 
rise to radiation of combination frequencies v v m . On the 
other hand, the Bohr theory postulates definitely that radiation 
is a mechanism in which the frequencies of the orbital motions 
of radiating electrons are not the frequencies of the spectral 
lines. These latter are, instead, proportional to the energy 
differences between the so-called stationary states in the mole- 



296 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XV 

cule. Moreover, in the Raman effect the intensities of the 
lines displaced toward the red are greater than those displaced 
toward the violet, as shown in Fig. 15-6, an effect also not in 
accordance with the classical concept of combination frequencies. 
The explanation of the Raman effect is really to be found as an 
integral part of the quantum theory of dispersion gind maybe 
reduced to the following simple terms.") 

Consider a molecule in the energy state indicated by a vibra- 
tional energy level a, Fig. 15-8, to be struck by a photon of 
energy hv, and raised to an energy state represented by level d, 
for which the transitions d > b or d c are not possible according 
to the selection rules of theory. Then Raman radiation is 
possible only if there exist in the molecule higher energy levels, 
represented by the group of horizontal dotted lines x, between 
which and the two levels 6 and c transitions are possible. It is 
to be understood that 6 and c likewise each represents a family 
of levels, so that groups of Raman lines will be observed. Also, 
the incident quantum may be that corresponding to any line 
emitted by the irradiating source. 

There are three possible ways in which radiation may take 
place. Either the molecule, upon being struck by a photon of 
energy hv, may scatter the same quantum, contributing to the 
intensity of a spectral line of the same wave-length as that of 
the incident photon (ordinary scattered light) ; it may reradiate 
a quantum & v\ = hv (E a E b ) where E a Eb is the difference 
of energy hAv between levels a and 6, contributing thereby to 
the intensity of a Raman line displaced toward the red; or it 
may reradiate a quantum hv 2 hv -f- (E c Ea), contributing 
to the intensity of a Raman line displaced toward the 
violet. The level a represents only one of a number of possible 
enerjjr states in which the molecule may be at the time it is 
struck by the photon. This bears upon the question of the 
intensities of the Stokes and anti-Stokes lines, and the dependence 
of these intensities upon the transition probabilities. If, as 
usually happens, the molecule is in a low energy state, represent- 
ing a relatively small total energy of the molecule, then the 
probability is enhanced that it will reradiate with energy 



h(v - 
If, on the other hand, the original level a is relatively high (a 



SEC. 15-19] LIGHT AND MATERIAL MEDIA 297 

more unusual circumstance for substances under ordinary 
temperature conditions), the probability is enhanced that it 
will reradiate with energy h(v + AJ>). Hence it is apparent 
that there is no dependence of the Raman line intensities upon the 
probability of transitions between levels a and b or a and c, but 
only on the probability of the transitions a * x and x b or 
x > c. 

{in the complete theory of the Raman effect/^of which the 
foregoing is only 'a~ 'C'dTTfteTfised and oversimplified account^ it is 
supposed that the transitions to and from the level represented 
by the dotted lines x are not real but virtual. This means 
that the initial photon of energy hv does not actually raise the 
molecular energy to the level x. If scattering takes place, the 
upper level is one such as d from which an actual transition 
d a may occur. If Raman lines appear, the dual energy 
change a > x and x > 6 theoretically represented as responsible 
for each Stokes line really consists of only a single transition, 1 
the same being true for the anti-Stokes lines. This theoretical 
interpretation agrees with the fact mentioned earlier, that in 
some cases no infrared absorption bands are found at frequencies 
corresponding to the values of Ai>. Not only do the differences 
of frequency Av appear in the Raman spectrum, but theory holds 
that the Raman lines cannot occur unless energy levels such as 
a and b actually existj In this manner the ftlaman effect 
offers an experimental method of finding those characteristic 
energy states of the molecule, even though there can be found 
no absorption bands in the spectrum to correspond to themj 

A superficial comparison or the Raman effect with fluorescence 
may leave the reader in doubt as to the difference between them, 
since in both cases the substance radiates energy corresponding 
to frequencies other than those of the irradiating light. In the 
case of fluorescence the reradiated energy is of a frequency 
which the fluorescing substance is able to absorb, with no depend- 
ence upon the frequency of the incident light, while in the 
Raman effect there is a fixed frequency difference Ai> between 
the displaced radiation and the incident radiation, no matter 
what the frequency of the latter may be. 2 

1 There will, accordingly, be a modification of the usual selection rules, 
given in Sec. 14-17. 

2 A full discussion of experimental work in the Raman effect will be found 
in R. W. Wood, "Physical Optics," Mapmillnn, 1934. 



298 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XV 

One important difference between ordinary and modified 
scattered radiation, i.e., between that which gives rise to the 
undisplaced spectrum line and that which causes the Raman 
lines, is in the phase relationship which the two bear to the 
incident light. In ordinary scattering there is a definite phase 
relation between the incident and scattered radiation. In the 
Raman effect the radiations from different molecules have phase 
differences which vary from one molecule to the next, and also 
different states of polarization. For this reason, ordinary 
scattering is called coherent, and the Raman, incoherent sc&tterine 

20. The Photoelectric Effect. For the most part the phenom- 
ena described in this chapter illustrate the importance of the 
quantum theory of radiation whenever the interaction of that 
radiation with matter is involved. The usefulness of that theory 
will be still more fully brought out in the following chapter. 
Historically, however, its first great success in explaining the 
interaction of light and matter was in connection with a phenom- 
enon which is not strictly optical, but which involves the effect 
of light, called the photoelectric effect. It is that whenever a 
metallic surface is irradiated by visible or ultraviolet light, 
x-rays, or -y-rays from radioactive substances, electrons, are 
ejected from the surface. The effect is much greater for some 
metals, such as sodium, potassium, and cesium, than for others, 
these metals being largely used, accordingly, in the construction 
of the modern photoelectric cell. It is found that the velocity 
possessed by an ejected electron depends, not upon the intensity 
of the radiation, but upon its frequency. This result cannot be 
explained on the basis of classical theory, since, if we call the 
kinetic energy of the electron )^wv 2 , there is every reason to 
believe that more intense radiation, possessing greater energy, 
might communicate more energy to the electron and thus give 
it a greater velocity than does weaker radiation. 

The true explanation was given by Einstein in 1905. By an 
extension of Planck's hypothesis, that the energy of the " oscil- 
lators" in a black-body radiation consists of integral amounts of 
some indivisible unit of energy e (see Sec. 14-13), proportional 
to hv. It follows inescapably from that theory that the energy 
of radiation must itself be quantized. 

Einstein carried this result still farther by the hypothesis that 
the energy of each quantum of radiation is not, as required by 



CHAPTER XVII 

THE EYE AND COLOR VISION 

The beginner or casual worker in the field of light is likely to 
overlook the importance of the eye in visual observations. It is 
important (a) because of considerations of purely geometrical 
optics, including defects of image formation; (6) because it has 
certain characteristics which may be classified as psychophysio- 
logical, such as susceptibility to illusions, color vision, and 




Fi. 17-1. The schematic eye. A, fovea; B, blind spot; C, cornea; D, 
aqueous humor, index = 1.3365; L, crystalline lens, index = 1.4371; E, vitreous 
humor, index = 1.3365; F, principal focal point. Radius of curvature of 
cornea, 7.829 mm.; of front of lens, 10,000 mm.; of rear of lens, 6.000 mm.; 
distance between cornea and lens = 3.6 mm.; distance between surfaces of 
lens = 3.6 mm. 

difference in degree of ''normality." Because of these, modifica- 
tion of observed phenomena is possible, and ignorance of this 
modification may lead the observer to false conclusions. Optical 
experiments, especially those involving visual photometry and 
color, should not be undertaken without some understanding 
of the functions of the human eye. 

1. The Optical System of the Eye. The essential optical 
features are illustrated in Fig. 17-1. The meanings of the letters 
are given in the legend. The surfaces here represented are not 
such definite boundaries between media as in ordinary optical 
systems. Neither are the media themselves entirely homogene- 
ous, the crystalline lens especially being composed of "shells" 

323 



324 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XVII 

which vary in density and structure. For these reasons it is 
customary in describing the optics of the eye to give the radii 
of curvature, indices of refraction, and other details of a "sche- 
matic eye" which in operation most closely duplicates the human 
eye. The diagram in Fig. 17-1 is that of a schematic eye. 

The portion of the retina where vision is most distinct is the 
fovea. The diameter of the fovea is about 0.25 mm. and sub- 
tends an angle somewhat less than one degree in the object space. 
The sensitivity of the retina diminishes with increasing distance 
from the fovea and the field of distinct vision is quite small. 
When one " looks at" an object, its image falls on the fovea. 
At the point where the optic nerve enters the eye, the retina is 
insensitive to light and is called the blind spot. The blind spot 




B 



FIG. 17-2. 



is a short distance from the fovea toward the nasal side, so that 
with either eye an object to one side of that on which attention 
is fixed may be unseen, provided it is the proper distance away. 
If the reader closes the right eye while Fig. 17-2 is held an 
appropriate distance away (about 6 in.), spot A will disappear 
when the attention is fixed on B, while with the left eye closed 
spot B will disappear when the attention is fixed on A. It may 
be necessary to experiment a little to find the proper distance 
of the book before this effect is obtained. 

2. Defects in the Optics of the Eye. Accommodation of the 
eye for objects at different distances is brought about by changes 
in the tension of the ciliary muscles which control the shape 
of the crystalline lens. 

The nearest position to the eye at which a small object can be 
distinctly seen is called the near point; that on which the eye is 
focused when relaxed, the far point of the eye. For nearby 
objects the lens is permitted to become more spherical in form, 
so that the focal length of the system is reduced. The power of 
accommodation decreases with age, so that it becomes difficult 
to distinguish small objects within the range of normal reading 
distance without the aid of glasses. There are also defects of the 
eye, not necessarily associated with age, which may be partly 
overcome with glasses. The three most common are myopia. 



SEC. 17-4] 



THE EYE AND COLOR VISION 



325 




hyperopia, and astigmatism. The first two are the result of 
abnormalities in the distance from the front of the eye to the 
retina, while the last is caused by lack of sphericity of the refract- 
ing surfaces, principally the cornea. An eye in which light 
from a distant object is focused exactly at the retina when 
accommodation is entirely relaxed is said to be emmetropic. 
Eyes which are myopic or hyperopic are said to be ametropic. 
These conditions are illustrated in Fig. 17-3. That the eye also 
suffers from barrel-shaped distortion 
can be shown by looking at a grid of 
perpendicular lines. The pattern will < a ) 
appear convex if held close to the 
oye. 

3. Binocular Vision. Ability to 
bring the image of an object simul- 
taneously on the fovea of each eye is 
called binocular vision. The pupils of 
the eyes in humans are separated by 
a distance of a few inches, so that 
with one eye the superposition of 
objects along the line of sight is not 
quite the same as it is with the other. 
The resulting slight difference in the 
images formed on the fovea of each 

eye enables one to determine depth in the object, or, in 
other words, to perceive space in three dimensions. Other 
factors enter into the situation, especially when illumination is 
poor, .the distance great, or the scene unfamiliar. A person 
having only one eye capable of seeing may make use "of other 
criteria of distance, such as the relative size of objects, their 
relative displacement in the case of motion, or a recollection of 
past experiences. 

4. The Stereoscope. In an ordinary photograph, objects at 
different distances are all projected on a single plane, so that the 
picture itself gives no effect of depth or relative distance and 
dependence must be made upon experience and judgment in 
forming a mental picture of the depth of the scene. To enhance 
the effect of depth, the stereoscope is used. Two photographs 
are taken, with a slight lateral displacement of the camera, or 
with a stereoscopic camera which takes two pictures at the same 




FIG. 17-3. 



326 



LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XVII 



B 




time with twin lens systems separated by a few inches. The 
prints are then mounted side by side and looked at through a 

stereoscope, one form of which is 
illustrated in Fig. 17-4. With 
a little practice some persons 
are able to achieve stereoscopic 
vision of a pair of photographs 
without aid, the process con- 
sisting of seeing each photo- 
graph separately, the left-hand 
picture with the left eye and 
the right-hand picture with the 
right eye, and bringing the two 
into coincidence. 
The principles of binocular vision are made use of in the con- 
struction of microscopes and telescopes, duplex optical systems 
being set side by side in the instrument. Some so-called binocular 
microscopes are not stereoscopic in the true sense, having merely 
two oculars both of which receive the image formed by the 
objective, through a mirror or prism system. The purpose in this 
case is to enable the observer to use both eyes and relieve 
eyestrain. 




Fio. 17-4.- A form of stereoscope. 



(a) 





Fio. 17-5. Optical illusions. 

5. Optical Illusions. Ocular experience with the common- 
place often leads one astray in viewing the uncommon. Ordi- 
nary optical illusions are illustrated in Fig. 17-5, in which a, 6, 



SEC. 17-71 THE EYE AND COLOR VISION 327 

and c are geometrical-optical, and d is due to irradiation. In d 
the white center circle looks larger than the black, although it is 
exactly the same size. Irradiation is sometimes a source of 
error in the measurement of spectral line positions, especially in 
absorption spectra. When the background between the lines 
is more dense than that on either side there is a tendency to 
estimate the lines to be farther apart than their positions shown 
by a purely objective microphotometric measurement. 

6. The Contrast Sensitivity of the Eye. While an extensive 
treatment of the limitations and capabilities of the human eye 
would carry us beyond the field of physics and into those of 
psychology and physiology, 1 certain characteristics of vision 
which are important in experimental optics will be discussed 
briefly in this and the following sections. 

The eye is designed to afford satisfactory vision over as wide 
a range of conditions as possible, and for this reason it is not a 
good judge of differences of brightness or intensity except under 
the most restricted conditions. The ability to distinguish 
between areas of different brightness is made use of in photom- 
etry. Most photometers are arranged so that the two fields to 
be compared are seen at the same time, one, the standard, being 
capable of fairly rapid variation of brightness. It is important 
that the two areas be arranged so that the effect of contour on 
relative brightness is reduced to a minimum. Ordinarily one 
of the areas is a small square or circle at the center of a like 
figure of considerably greater area. The contrast sensitivity may 
be measured by adjusting the brightness of the center spot so 
that it is barely different from that of the larger area. If the 
difference of brightness is A#, and the brightness of the larger 
area B, then AB/B is the contrast sensitivity. It is practically 
constant for brightness above about 1 candle per square meter, 
but it increases very rapidly as the brightness decreases. 

7. Flicker Sensitivity. Persistence of Vision. The sensation 
in the retina does not cease at once when the stimulus is removed, 
and in consequence the intermittency of a flickering light will 
not be detected, provided the flicker is rapid enough. With a 

1 See, for instance, Helmholtz, "Physiological Optics," English trans- 
lation by J. P. C. Southall, published by the Optical Society of America; 
also Troland, "Psycho-physiology," Van Nostrand; Parsons, "Introduction 
to the Study of Colour Vision," Cambridge University Press; and Collins, 
"Colour Blindness," Harcourt, Brace. 



328 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XVII 

field brightness of about 1 candle per square meter the critical 
frequency beyond which no flicker may be detected is about 
30 times per second. The critical frequency is a function of the 
alternation in brightness. 

A flicker method is often used for the comparison of photo- 
graphs in which slight changes are to be sought, as in photographs 
of areas of the sky, taken at different times. In order to detect 
the presence of stellar objects whose proper motion (motion 
across the line of sight) is great compared to those of the general 
background of stars, the two photographs are arranged so that 
by shifting a mirror back and forth, first one and then the other 
may be seen in the field of a microscope. If any stellar object 
is in different positions in the two photographs, its displacement 
relative to the general background may be detected and, with a 
micrometer eyepiece, measured. 

The flicker photometer may be used for the comparison of the 
intensities of two sources between which there is a considerable 
difference of color. The light of one source is reflected to the 
eye from a stationary white screen Si. The light of the other 
source is reflected from the surface of a rotating disk 82, with 
white vanes. This disk is so placed that Si may be seen through 
its open spaces, which have the same total area as the vanes. 
The disk may be rotated at a speed such that while the colors 
blend, the illuminations do not. The sense of flicker which is 
experienced when the two sources are not of the same intensity 
disappears when their distances arc adjusted so as to equalize 
the illuminations. The flicker photometer should be used under 
carefully controlled conditions, and only when the intensities are 
sufficiently high so that there is no Purkinje effect (see Sec. 17-8). 

Flicker methods are also used to reduce the intensity of a 
source, the light usually being passed through a disk from which 
sectors have been cut. In this case the rate at which the light 
is alternated by reason of the interposition of the opaque parts 
of the disk must be greater than the critical frequency mentioned 
above. It has been proved that the apparent brightness of an 
object viewed through such a rotating disk is proportional to the 
ratio of the angular aperture of the open to the opaque sectors. 
This is known as Talbot's law. 

8. Spectral Sensitivity. The sensitivity of the normal eye 
as a function of wave-length is shown by the solid curve in Fig. 17-6 



SEC. 17-9] 



THE EYE AND COLOR VISION 



329 



for ordinary illumination. For illumination at the threshold of 
vision the maximum of visibility shows a marked shift to the 
violet as given by the dotted curve. Both curves are plotted 
with relative visibility as ordinate in arbitrary units. This shift 
of the wave-length region of maximum visibility is called the 
Purkinje effect, after its discoverer. It is generally attributed 
to the character of the adaptation which the eye undergoes at 
low intensities of illumination. This "darkness adaptation" is 
an increase of acuity of vision for brightness but not for color. l 

9. Color. In the field of physics an object is said to have a 
given surface color when it exhibits a certain selective absorption. 




4000 



7000 



5000 6000 

Angstroms 

FIG. 17-6. The Purkinje Effect. Solid curve shoVs relative visibility for ordi- 
nary brightnesses; dotted curve, at threshold of vision, on an arbitrary scale. 

There is a household usage of the term color characterized by its 
association with the words tint, a mixture of a color with white, 
and shade, a mixture with black. In the field of color vision 
still a third meaning is introduced, that used by the psychologist 
and physiologist in referring to a given sensation transmitted 
by the eye as a result of an external physical stimulus. More- 
over, the term spectrum has a different significance in different 
fields. The physicist thinks of the spectrum of visible light as 
a wave-length band terminating in long waves associated with 
deep red fading into invisible infrared at one end, and in violet 
fading into invisible ultraviolet at the other. On the other hand, 
to the psychologist the colors form a continuous circle, the violet 
being a blend of red and blue in which blue predominates, and, 

1 "In the dark all cats are gray." Old proverb. 



330 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XVII 

beyond the violet, purple, a "nonspectral" blend of blue and red 
in which red predominates once more. 1 

The psychological definition of color is perhaps best given in 
the following words: 2 " Color is the general name for all sensations 
arising from the activity of the retina of the eye and its attached 
nervous mechanisms, this activity being, in nearly every case 
in the normal individual, a specific response to radiant energy of 
certain wave-lengths and intensities." 

10. Hue. The spectrum is said to be made up of hues. 
Four of these, red, yellow, green, and blue, are unique in that 
they are not composed of mixtures of others. Orange is con- 
sidered as a mixture of red and yellow. Two other blends of 
contiguous hues are blue-green and yellow-green. Violet is a 
mixture of red and blue with blue predominating; purple, a 
nonspectral mixture of red and blue with red predominating. 
With the addition of black and white, from these nine hues all 
colors may be produced. "Hue is that attribute of certain 
colors in respect of which they differ characteristically from the 
gray of the same brilliance and which permits them to be classed 
as reddish, yellowish, greenish, or bluish." 

The sensation of white is produced by any color if sufficiently 
intense. Hence yellow, which produces relatively the largest 
stimulus, is said to contain the greatest amount of white. 

11. Saturation. A color is said to be saturated when it is 
mixed with the smallest possible quantity of white or black. 
According to the preceding section, yellow is less saturated than 
the red obtained from the same white-light spectrum. However, 
if the entire spectrum is reduced in luminosity, the red is said 
to become desaturated by mixture with black, while at the same 
time the yellow approaches saturation by a reduction of its 

1 As a result of the combined planning and research of those whose chief 
interest is in the field of colorimetry, the subject of color has been lifted from 
the realm of vague concept and discordant terminology to the position of a 
well-developed technology with precise techniques. This has come about 
largely through the exchange of ideas and deliberations of international 
commissions meeting at intervals of several years, and dealing with the 
subjects of illumination, color, and spectrophotometry. Several of the 
references in the following sections are to reports of these commissions. 

2 This definition, as well as those of hue, saturation, and brilliance quoted 
in following sections, are from the Report of the Committee on Colorimetry 
for 1920-1921, Jour. Opt. Soc. Amer., 6, 527. 



SEC. 17-13] THE EYE AND COLOR VISION 331 

luminosity. "Saturation is that attribute of all colors possessing 
a hue which determines their degree of difference from a gray 
of the same brilliance." 

12. Brilliance. The term closest to this in meaning in physics 
is brightness'or luminosity, but since these have already been 
used with objective meaning, the term brilliance will be used to 
indicate the relative excitability of the retina for different parts 
of the spectrum. Thus, the yellow is the most brilliant color 
in the spectrum of a white-light source of ordinary intensity. 1 
"Brilliance is that attribute of any color in respect of which it 
may be classed as equivalent to some member of a series of grays 
ranging between black and white." 

13. Color and the Retina. The retina of the human eye is a 
complicated structure composed of many layers, each of a 
composite structure. The parts most directly associated in 
theory with color vision are the rods and cones. That the rods 
and cones play an important part in the mechanism is shown by 
the following observed relations. 2 

a. In case of congenital absence of both rods and cones, blindness 
exists. 

6. If the fovea has no rods, that part of the retina suffers from night 
blindness, a term describing various degrees of inability to see with low 
illumination. 

c. Color blindness accompanies a congenital absence of cones. 

d. Animals having a predominance of rods (bats, owls, etc.) have good 
night vision and poor day vision, while birds, with a predominance of 
cones, have the opposite characteristics. 

e. Rapidity of adaptation to dark is associated with the extent of 
changes which take place in the rods. 

The relationships just given support the theory that the rods 
are important in brightness vision and the cones in color vision. 

All parts of the retina do not have the same degree of sensitivity 
to color, which is probably due to the cones becoming relatively 
infrequent as the periphery is reached. In normal eyes the 
retina is sensitive to yellow over the largest area and to blue 

1 It is perhaps worth while to warn against confusion of this distribution 
of brilliance with the distribution of radiant energy associated with a 
source at a given temperature, as described by Wien's distribution law. 

2 These relations have been adapted from Bills, "General Experimental 
Psychology," Ixmgmans. 



332 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XVII 

over one almovst as large, to red over a still smaller area, and to 
green over the smallest. 

14. Complementary Colors. If two colored lights are mixed, 
the resulting stimulus matches that of a third, the exact color 
of which depends upon the proportions of the mixture. Often in 
such cases the match is not perfect, the mixture being less 
saturated than the third color. As the spectral separation 
(difference of wave-length) of the two colors mixed is increased, 
the saturation becomes less. Two colors sufficiently far apart 
in the spectrum give, when mixed, the sensation of white. Such 
colors are called complementary. Table 17-1, of complementary 
colors, is due to Helmholtz. 1 



Color 

Red 6562A 

Orange \ 6077A 



Yellow, 

Yellow 

Yellow 

Yellow 

Yellow green 



5853A 
5739A 
5671A 
5644A 
5636A 



Complementary color 



Green blue I 4921 A 

Blue i 4897A 

Blue j 4851A 

Blue I 4821A 

Indigo blue 4G45A 

Indigo blue 461 8A 

Violet j 4330 A and beyond 



16. Theories of Color Vision. 2 It is found by experiment that 
a color stimulus may be accurately matched by a mixture of 
correct amounts of three color stimuli. The first person to make 
use of this as the basis of a mechanistic theory of color vision 
seems to have been the versatile genius, Thomas Young. His 
postulation of the existence in the human eye of three independ- 
ent mechanisms of color perception, each correlated with one 
of the three primaries, red, green, and blue, is the basis of what 
is now universally known as the Young-Helmholtz theory of 
color vision. Equal stimulation of all three mechanisms results 

1 A more extended table of complementaries, based on the standard 
source for colorimetry, used in place of the so-called white-light source 
of earlier research (see Sec. 17-18), is to be found on p. 31 of the "Handbook 
of Colorimetry," by A. C. Hardy, The Technological Press, 1936. The 
values of complementaries listed in this table are those of the dominant 
wave-lengths of complementary colors (see item 6, Sec. 17-19). 

2 For a more extensive treatment see Parsons, "An Introduction to the 
Study of Colour Vision," Cambridge University Press. 



SBC. 17-15] THE EYE AND COLOR VISION 333 

in gray. Black is the absence of any stimulus. The relative 
sensitivity of the different mechanisms is illustrated by Fig. 17-7. 
The Young-Helmholtz theory accounts for after-images as due 
to retinal fatigue, but does not account for the gray after-image 
of black nor the black after-image of gray. It does riot account 
for contrast, nor for the existence of color-sensitive zones in the 
retina. It accounts only partly for color blindness, not providing 
for the gray vision of the color blind. On the other hand, the 
correspondence between the fundamental postulate of Young and 
the experimental facts of the science of colorimetry make the 
theory a suitable conveyance for the concepts and nomenclature 
of the purely metrical phases of that science. 




R 



4000 5000 6000 " "" " 7000 

FIG. 17-7. Relative sensitivity of the red, green, and blue mechanisms of 
color perception. The shape of the curves is illustrative only, and conforms 
to no particular set of data. 

The theory of Bering claims the existence in the retina of two 
mutually exclusive processes: (a) anabolism, the process by 
which matter is transformed into tissues; and (6) catabolism, the 
process by which substance is broken down in the tissue. This 
theory recognizes the presence in the retina of three mechanisms 
which can be excited in either of these processes. Anabolic 
excitation yields the sensations of green, blue, and black; cata- 
bolic, red, yellow, and white. This theory explains the phe- 
nomena of complementary colors, but not the mixture of black 
and white to form gray. It accounts only partly for color 
blindness. 

The theory of Ladd-Franklin assumes that in the rods and 
cones of the retina exist types of molecules which are affected 
and modified by the action of the light. This bold hypothesis 
goes far to bring the trichromatic theory of Young-Helmholtz 
and that of Hering into accord. It does not account for binoc- 



334 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XVII 

ular effects. Also, the possibility of molecular changes and 
motions occurring with the rapidity required by visual phe- 
nomena has been gravely questioned. It is a theory which 
concerns chiefly the psychologist and physiologist. Those 
interested only in the physical aspects of color vision and colori- 
metric measurements find the trichromatic theory of Young a 
suitable conveyance for the concepts and definitions involved in 
their work. 

16. Color Mixing versus Pigment Mixing. Colored lights may 
be mixed in a variety of ways, some of which will be described 
in the next section. A simple method, however, is to paint 
on a disk a red sector and a green one. With a suitable choice 
of angle of the two sectors, on rotating the disk the visual sensa- 
tion will be yellow. If some of the same pigments are mixed, the 
mixture will not appear yellow, but dull brown. The difference 
is that while in the first case there is a true mixture of the two 
stimuli, both occurring at the retina, in the second case the light 
received by the eye is that which is not absorbed. With the 
red and green pigments mixed, the light which is not entirely 
absorbed contains not only yellow, but some red and some green. 

17. Colorimeters. A colorimeter is an instrument for measur- 
ing the character and intensity of a stimulus due to a color or a 
mixture of colors. One of the earliest precision colorimeters is 
the color-patch colorimeter of Abney. This is a spectrometric 
device equipped with two or more slits at the plane where the 
spectrum is focused, by means of which varying relative amounts 
of different spectral regions can be isolated. These are then 
brought into superposition in a field of some area and compared 
with the original white light. Another instrument, designed by 
H. E. Ives, makes use of filters instead of slits to isolate the 
spectral primaries. There are many difficulties to be overcome 
in the construction and use of a colorimeter, in part because of 
the dual character of vision, i.e., sensitivity to color and to 
brilliance. While the subject is too extensive for complete 
treatment here, certain developments of the past decade which 
have transformed colorimetry into a precise quantitative science 
will be discussed. 

18. Color Mixing. It is found by experiment that a color 
stimulus may bo accurately matched by a mixture of correct 
amounts of three color stimuli. Three colors thus used are 



SEC. 17-18] THE EYE AND COLOR VISION 335 

called primaries. No three primaries will combine to match all 
colors, but, as will be seen later, this is not as severe a limitation 
as might at first appear. We may express this additive char- 
acter of color stimuli by the equation 

S = Pi + P 2 + P 3 , (17-1) 

where S is the color stimulus to be matched and PI, PI, and PS, 
are the three primaries. Sometimes the color stimulus produced 
by the mixture is unsaturated, and to compensate for this a 
suitable amount of white must be added to 8. 

In the earlier work done in color mixture, the different regions 
of the spectrum were matched with combinations of three given 
primaries, and the amounts of the primaries recorded by the 
observer. But observers differ slightly among themselves, even 
though they have normal color vision. Consequently, in more 
recent compilations of colorimetric data it has been the practice 
to average the results obtained by numbers of carefully selected 
observers. Those data have been standardized by international 
commissions. The negative values of the primary stimuli which 
occur in matching certain spectral colors with any given set of 
primaries are eliminated by a simple mathematical transforma- 
tion. Let r, g, and b be three values in energy units of the three 
original primaries which, an observer finds, will mix to match 
a certain wave-length from a given source, arid r', g f , and b f the 
translated values in terms of a new set of primaries. Then 

r' = kir + fag 4- fab,} 

g' = far + fag + fab f > (17-2) 

6' = far + fag + fab,j 

where the k's are the values of the original primaries in terms of 
the new primaries. Thus the values obtained with any set of 
primaries may be translated in terms of any other set, and hence 
in terms of a set so chosen that it contains no negative values. 
It follows, however, that the set so chosen by international 
agreement is based on primaries which are not real colors, an 
expedient which, because of the linear transformation given 
above, causes no unsurmountable difficulty. 

The values of the primaries corresponding to wave-lengths 
at intervals of 50 angstroms throughout the visible spectrum 
are given in energy units in the report of the Committee on 



336 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XVII 



Colorimetry 1 for 1928-1931, and also in the "Handbook of 
Colorimetry." 2 These values are called tristimulus values and 
are designated by x, y, z for luminous sources and X, Y, Z for 

diffuse reflection from colored 
surfaces. In Fig. 17-8 the values 
are plotted as ordinates against 
wave-length as abscissas. For 
instance, the tristimulus values 
of the recommended standard 
source (illuminant C, Appendix 
VII) for wave-length 4800 ang- 
stroms are given by the ordinates 
at that wave-length of the three 
curves. 

Thus it is now possible to obtain 
the chromaticity, or color value, of 
a sourco m terms of an interna- 

FIG. 17-8. Tristimulus values for .. i A i t -^ 

standard illuminant C. (Adapted tionally adopted set Ot specifica- 

from A. C. Hardy, "Handbook of tions by comparing it spectro- 
C'olorimetry."') . . 

photometrically with the adopted 

standard. The chromaticity is given in terms of three so-called 
trichromatic coefficients: 





4000 



5000 



X 

x = , _- 
x + y 



y - 
* x 



'$ + * 



Z - -: . . 

x 4- y + 



(17-3) 



The chromaticity is by this means evaluated as a quantity 
independent of the total brightness (brilliance). 

19. Graphical Representations of Chromaticity. a. The Color 
Triangle. The experimental results of color mixture give support 
to the construction of a geometrical figure which will express 
graphically all the known results and concepts associated with 
the science of colorimetry. If the concept of brilliance is omitted, 
this can be done on a plane figure called the color triangle, shown 

1 Transactions of the Optical Society (London), 33, 73, 1931-1932. 

2 Compiled by A. C. Hardy and associates; published by the Technology 
Press, 1936. 



SE<\ 17-19] 



THE EYE AND COLOR VISION 



337 



Green 



in Fig. 17-9. The color triangle proper is shown by the heavy 
inscribed line along which the respective spectral positions are 
given by the Fraunhofer letters. 

In order to express also the concept of brilliance, the color 

diagram must have a third 
dimension. The resulting fig- 
ure is generally called a color 
pyramid. No single three- 
dimensional diagram has been 
proposed which embodies all 
the experimental facts and the 
concepts of color vision. Per- 
haps the best figure is one 
which indicates only the dimen- 
-g fue sions as in Fig. 17-10. l In 
T.- TT n TU i * 'i what follows, however, it will 

FIG. 17-9. The color triangle. ' ' 

be seen that in reality only two 

dimensions are required, provided the tristimulus values are 
evaluated in terms of trichromatic coefficients. 

White 




D 






Blue 



Green 



te/Aj>w 



Black 
FIG. 17-10. A three-dimensional color diagram. 

b. The Chromaticity Diagram. For purposes of colorimetric 
evaluation, the color triangle has been standardized, and by 
international commission has been referred, not to the indefinite 
quantity known as white light, but to standard illuminant C 
(see Appendix VII). The resulting figure is called a chromaticity 

1 Adapted from the report of the Committee on Colorimetry, Journal of 
the Optical Society of America, 6, 527, 1922. 



338 LIGHT: PRINCIPLES AND EXPERIMENTS [CHAP. XVII 

diagram, shown in Fig. 17-11. 1 The dotted line joining the ends 
is the region in which the nonspectral color mixtures (purple) are 
located. The saturation of each color is given by its distance 
from the center point, which represents white. A straight line 
drawn through the white point terminates in two colors which 
are complementary. The coordinates of each point on the 
curved line are the trichromatic coefficients of a wave-length 




0.1 0.2 0.3 0.4 0.5 0.6 0.7 

FIG. 17-11. A chromaticity diagram. The numbers on the curve indicate 
the wave-lengths in millimicrons (1 millimicron = 10 angstroms). (Adapted 
from A. C. Hardy, "Handbook of Colorimetry") 

in the spectrum between 4000 and 7000 angstroms, calculated 
from the tristimulus values for the standard illuminant. The 
chromaticity of any source of light or colored surface is given 
by a point in the diagram. For instance the chromaticity of the 
standard illuminant C is given by the point C at the approximate 
coordinate values x 0.310, y 0.316. Several interesting 
properties are given by the chromaticity diagram : 

a. The color resulting from a mixture of two colors, say, red and green, 
will lie on the straight line joining their chromaticity points, R and 0. 

1 Adapted from A. C. Hardy, "Handbook of Colorimetry," The Technology 
Press, 1936. 



SEC. 17-19] THE EYE AND COLOR VISION 339 

b. Hence, if a straight line is drawn from C to a point I) on the curve 
(see Fig. 17-11), the color corresponding to any point on that straight 
line will result from a mixture of illuminant C and the spectrum color 
corresponding to /). The point D then gives the dominant wave-length 
of the color in question. 

c. The purples are all represented by points lying within the dotted 
triangle. 

d. As in the case of the earlier color triangles, complementaries as, 
for instance, G and P lie on a straight line drawn through C, which is 
analogous to the white point. 

Problems 

1. An object is 30 cm. from the eye. What is the numerical aperture 
when the entrance pupil of the eye is 5 mm.? 

2. A person whose vision is hypermetropic possesses a range of 
accommodation permitting him to see clearly objects closer than 150 cm. 
If he is fitted with glasses which are convergent lenses of 20 cm. focal 
length, how near may he bring a printed page and still see the print 
clearly? 

3. A farsighted person can see objects clearly if they are more than 
50 cm. away. If he uses a reading glass of 15 cm. focal length, what 
lateral magnification does he obtain? 

4. A person with normal vision adjusts a telescope for his own use. 
It is then used by a person who has no power of accomodation for nearby 
objects. What adjustments should the second person make? If it is 
to be used instead by a person who is very shortsighted, what adjust- 
ments should he make? 

5. Can objects bo seen distinctly when the eye and object are under 
water? Explain your answer. 



EXPERIMENTS IN LIGHT 



EXPERIMENT 1 
FOCAL LENGTHS OF SIMPLE LENSES 

Apparatus. An optical bench about 2 m. long; an assortment 
of convergent arid divergent lenses; a source of light; a glass 
mirror which can be rotated about horizontal and vertical axes; 
a ground glass or white screen; a spherometer ; calipers; meter 
sticks, steel tape. The source of light may be a frosted electric 
light bulb enclosed in a metal box which has one side plane and 
painted white, with an opening in the white area crossed by wires. 
If the lenses are thick, it is desirable that they should be mounted 
in metal cells on which are marked rings to indicate the principal 
planes. The distance which ordinarily would be measured to a 
thin lens should then be measured to the appropriate principal 
plane of the lens. 

OB A I 





/ 


v / 


\ 








* i 


\ 




Source 


/ 


\ i 


\ 


Screen 




/ 


\ ' 


\ 

i 






i 


; ' i 


i 






\ 


/ \ 


/ 






\ 


/ \ 


/ 






M H 




^ . _ . a >j i< - - ~-- ~-~a *- - -Vr_ in irr^v.^j 



Fia. 1. If a + a' is greater than 4/, there will be two positions of the lens for 

which a focus is obtained. 

Part A. The Focal Length of a Simple Lens. Set up the 

source at one end of the optical bench and the white screen at the 
other. Select a double-convex lens whose focal length, roughly 
determined by obtaining the image of a distant object, is between 
20 and 30 cm. It may be as much as 40 to 45 cm., but a shorter 
length is desirable. A plano-convex lens may be used, in which 
case the convex side should be toward the source, since in this 
position the spherical aberration of such a lens is a minimum. 
Set the lens with its axis parallel to the bench, and slide it along 
until an image is formed on the screen. If the image distance 
from the lens is smaller than the object distance, as for position A 
(Fig. 1), there will be another lens position at B for which there 

343 



344 



LIGHT: PRINCIPLES AND EXPERIMENTS 



will bo an image on the screen. Then, referring to Fig. 1, 
0,2 = ai and 2 = a\. Measure the distances as accurately as 
possible and calculate the focal length / of the lens from the 

equation | 7 = -v Keep in mind that unless the lens is 
a a' f 

thin, the values of a and a' should be measured from the object 
and image planes to the principal planes P and P', respectively. 



Source 



Image - 



Lens 

r\ 






Pfane 
Mirror 




FIG. 2. Auto-colli mating method for determining/. 

Part B. Focal Length by the Autocollimation Method. Set 

up the source, lens, and plane mirror as shown in Fig. 2, using 
the same lens as in Part A. Adjust the mirror and lens so that 
an image of the cross wires falls on the white surface of the lamp 

enclosure. The distance from the lens 
(or its nearer principal plane if this is 
known) to the cross wires is the prin- 
cipal focal length /. 

Part C. Index of Refraction with a 
Spherometer. First set the sphero- 
meter (Fig. 3) on a plane glass surface 
or metal plate and screw the center 
point up or down until it is just in 
contact with the plate. If it is too far 
down, the spherometer will rock on its 
legs. Holding the center knob firmly 
in this position, twist the micrometer 
dial on its shaft until its zero mark 
comes into coincidence with the vertical 




V 



Fio. 3. A Bpherometer. 



scale. The reading on the vertical scale is the zero mark 
for the measurement which is to be made, and should be 
on one of the divisions. If it is not, it is probable that the 
micrometer either is not flat or its plane is not perpendic- 
ular to the screw of the spherometer, in which case a record 



EXP. 1J 



FOCAL LENGTHS OF SIMPLE LENSES 



345 



of the variation should be made. Place the spherometer on the 
lens surface to be measured and turn the center knob until the 
center point just makes contact with the lens with no rocking. 
Record the amount the center point has been elevated (or 
depressed for a concave surface) from the zero point previously 
determined. Press the leg points on a piece of paper and measure 
the three distances d between each pair of legs, and obtain an 
average value for d. In case the points are flattened by wear, be 
sure that the distances measured are not to the centers of the 
depressions in the paper but to the edges corresponding to the 
points of the legs which were in contact with the lens surface. 
Calculate the radius of curvature by means of the equation 



_ 
r ~ 



d 2 
6i 



d) 



where s is the distance measured with the spherometer. Repeat 
for the second surface. Calculate the index of refraction by 
means of the equation 



1 , ,Jl l\ 

7 = (n - 1)1 I- 

/ Vi r t / 



(2) 



Part D. The Focal Length of a Divergent Lens. Choose a 
convergent lens of somewhat longer focal length than the diver- 
gent lens to be measured and set it up on the optical bench as in 




FIG. 4. 



Fig. 4. There will be a real image at /i. Place the divergent 
lens L 2 between this image and LI. Then /i will serve as a 
virtual source for which L 2 will form an image at 7 2 . The 
distances 7iL 2 and 7 2 L 2 are a and a', respectively, in the equation 



a 



a 



/ 



Part E. Index of Refraction of a Divergent Lens. Repeat 
Part C for the divergent lens whose focal length has been found 
in Part D. 



346 LIGHT: PRINCIPLES AND EXPERIMENTS 

PartE'. Curvature of a Concave Surface. Second Method. 

The curvature of the concave surface may be found by the sphe- 
rometer and also by another method which serves as a check on 
the spherometer measurement. In a suitable clamp set up a 
polished strip or small rod of metal, as for instance, a large 
needle. Illuminate it with a lamp held near by, and set it at a 
position in front of the concave surface where the inverted 
image of its point will coincide exactly with it as shown in Fig. 5. 
The point of coincidence may be determined by eliminating the 
parallactic displacement as the eye is moved from side to side 
and up and down. If the other lens surface also reflects enough 
light to interfere, smear it with a little vaseline which can be 



\Image 
i 

I ) i 

\ i ^~- 

fy e A A A / \J^*~* 



~N V , 

yeti 6 fl 



Eyepiece 



/ \ 



Object 



FIG. 6. A point and its image coincide at the center of curvature of a spherical 

mirror. 

wiped off later with a soft cloth or lens paper. Some observers 
will find it desirable to use an eyepiece of moderate power in 
eliminating the parallax. The distance from the object point 
to the lens surface is the radius of curvature of the surface. 

In addition to the details of measuring technique and manipula- 
tive skill in this experiment, there are some important lessons 
to be learned regarding the effect of inaccuracies in the different 
observations. For the divergent lens, calculate the error intro- 
duced into the measurement of radius of curvature by an error 
of 1 per cent in the measurement of the distance between the 
spherometer legs. How does this compare with the mean error 
of three successive observations of r by method E"! Conclude 
your report of the experiment with a discussion of the relative 
accuracy of measurement of / by the methods outlined, the 
probable percentage of error in the measurement of the index 
of refraction, and the sources of all errors. 

If the principal planes of the lenses are known, what additional 
accuracy is gained by measuring all distances from them rather 



EXP. 2] CARDINAL POINTS OF LENS SYSTEMS 347 

than from the vertices of the lens surfaces or the mean positions 
of the surfaces? 
Verify eq. 1. 

EXPERIMENT 2 

CARDINAL POINTS OF LENS SYSTEMS 

The theory of lens systems will be found in Chap. III. 

Apparatus. A source of light and a collimating lens for the 
projection of a parallel beam; a nodal slide; a small screen on 
which to focus images; an assortment of lenses. 

The source of light may be a concentrated point, or a tungsten 
lamp filament which lies nearly in a single plane. The collimator 
should be a fairly good lens of 10 to 15 cm. diameter and of 
about 1 m. focal length. It may be placed in one end of a tube 
or box, with the lamp at the other end, capable of adjustment 
for focal distance. The beam may be collimated with a labora- 
tory telescope, previously focused for parallel light, set up in the 
beam, with a smoked-glass filter between the eyepiece and the 
eye to prevent injury to the eye 1 . 

A nodal slide is essentially an optical bench which may be 
turned about a vertical axis. It is possible to obtain such 
apparatus possessing many 

refinements, but a satisfactory i > ; t. 
arrangement is that shown in ^ , ,\)J )/""" 

O ^Si 1 IT 1 L I I I II 1 1 I 1 1 1 I 1 I I jjl i i i i i j 

Fig. 1. A rigid bar of metal or S'''' 

wood is made in the shape of a 

trough in which may be placed 

the cylinders containing the FIG. I.--A nodal slide. A A, two 

loncna TTr Kor i rnmintoH lenses in cylindrical cells; B, a cylinder 
lenses. ne oar in inuuiiteti .,, i * o .. u u ^.u i o 

with a slot o to hold the lenses; o , a 

with One end clamped On a scale; T, & tripod support; /', a pointer 

spectrometer table, or on an at the axis of the slide, 
improvised vertical axis, and is equipped with a pointer P, as 
shown, by which the axis of rotation may be determined. A 
scale or meter stick should be fastened to the side of the bar. 

The screen may be a ground-glass disk about 1 in. in diameter, 
mounted on an arm which can slide along a bar parallel to the 
direction of the beam of light. 

The lenses need not be of the same diameter, but should 
be mounted in brass cylinders of the same size, on which are 



P 



_ 

( i i i j i i i i 




348 LIGHT: PRINCIPLES 'AND EXPERIMENTS 

marked rings indicating the positions of the principal planes. 
A tube about 8 in. long, with a 3^-in- slot cut nearly its whole 
length should be provided, into which the lens cylinders can 
be fitted, thus holding them a fixed distance apart for each set 
of measurements. 

The experiment is to determine the manner in which the 
cardinal points of a combination of two equal lenses vary in 
position as the distance d between them is changed. The 
theory and equations for the positions of the cardinal points will 
be found in Chap. III. 

Select two biconvex lenses of equal focal length, say, about 
15 to 20 cm. Place one of them on the nodal slide and measure 
its focal length, which will be the distance from an image of 
a distant source to the emergent principal plane P 9 . Repeat 
for the other lens. Put the two lenses together in the holding 
tube, with the distance d between their inner principal planes as 
small as possible. Lay the combination on the nodal slide, 
obtain a good focus of the source on the screen, and rotate the 
slide back and forth about its vertical axis. If the image also 
moves from side to side, move the lens combination and the 
screen together along the nodal slide, until a position is reached 
where rotation of the slide causes no shift of the image from side 
to side. It is essential to keep the image well focused as these 
maneuvers are carried out. When the position of no lateral 
shift is reached, the axis of rotation of the slide passes through the 
emergent nodal point of the combination. Since the system has 
the same medium on both sides, this is also the emergent principal 
point. 

Record the distances /', p', and d for about eight different 
separations of the lenses. The value of d should be varied from 
the smallest to the largest obtainable experimentally. Make a 
comparative table of these values and those calculated from 
equations. 

Discuss the reasons for the differences between the calculated 
and observed values. 



Exi>. 3] A STUDY OF ABERRATIONS 349 

EXPERIMENT 3 
A STUDY OF ABERRATIONS 

For the theory of Aberrations see Chap. VI. 

Apparatus. An optical bench, a mounting for lenses with a 
turntable graduated to degrees, a concentrated source of light, 
lenses, diaphragms, and red, blue, and green filters. 

Spherical Aberration. The source of light should preferably 
be monochromatic. This may be obtained by focusing the 
light of a mercury arc on a hole about 1 mm. in diameter through 
a color filter which transmits only the green line of mercury, 
5461 angstroms. If this is too faint, use a concentrated filament 
lamp or a Point-o-lite. Mount the turn- 
table so that it may slide along the bench. 
The screen may be a large sheet of bristol 
board, or white celluloid. 

For a lens, an ordinary projection 
lantern condenser is suitable. It should 
be mounted in a brass cylinder on which 
grooves are cut coinciding with the 
principal planes. A diaphragm like that 
shown in Fig. 1 is then mounted over the 
lens. Notice that the holes are arranged, 

not radially, but so that the images will not fall on each other 
as the focal position is changed. 

The purpose of the experiment is to obtain a set of measure- 
ments of the longitudinal spherical aberration (L.S.A.) for differ- 
ent focal lengths. Adjust the lens in its holder with the convex 
side toward the source, and with its axis passing through the 
source. For several object positions, each obtained by moving 
the lens, find the focal distance for rays through the center 
opening, and also for rays through the openings at distances 
hi t hz, and h^ from the center. The maximum range of difference 
of the focal distance between the focus for the center opening 
and that for the outermost zone should give the L. S. A. Calcu- 
late the values of the L. S. A. from the following equation: 

L . S . A . . "( ~ i) . *'[(! + 1Y . (..+_! + 1) 

n 2 2L\ri a/ \ a r l j 

I IV /n+l 1 

n ~ 7) ' \~Hr- 7 




350 



LIGHT: PRINCIPLES AND EXPERIMENTS 



If a plano-convex lens is used, with the plane surface away 
from the source, rz = , and h is the appropriate distance of a 
given circle of holes from the center of the diaphragm. 

Plot on a graph the L.S.A. for the outermost holes against the 
value of a', which is the image distance for the center hole. 
Write a brief statement regarding the use of a front stop to reduce 
aberration in the lens used. 

Astigmatism. To obtain astigmatic focal distances it is 
preferable to select a double-convex lens of about 10 cm. diam- 
eter and 30 to 40 cm. focal length. Prepare a diaphragm having 

two rows of holes, as shown in Fig. 2, 
arid place it over the lens so that one row 
of holes coincides with the axis of the 
turntable. Rotate the turntable and 
lens so that the optic axis makes an angle 
of about 20 deg. with the direction to 
the source. Move the screen so that the 
horizontal row of holes comes approxi- 
mately to a point or area of confusion. 
The distance from the lens to the screen 
is the primary focal distance si. The 
distance s* will be found by placing the screen similarly at the 
position where the vertical row of holes comes to a focus. Calcu- 
late the values of s\ and s% from the equations 




FIG. 2. 



2(n 



s 

- 

s 



s\ 



r cos i 
2(n 1) cos i 

. . - , . y 

r 



and the value of the astigmatic difference, s 2 i, from 

1 1 2(n 1) sin i - tan i 



Coma. Select a plano-convex lens, a double-convex lens, an 
ordinary achromatic doublet, and a spherical mirror. If pos- 
sible, they should have about the same relative aperture. Each 
lens should be divided into alternate transmitting and opaque 
concentric zones. It is easy to render the nontransmitting zones 
opaque by pasting on the lens rings cut out of black paper, but, if 
the lenses are needed for other purposes, these rings may be 



EXP. 3] A STUDY OF ABERRATIONS 351 

pasted on a thin disk of glass the size of the lens. A transmitting 
center disk and three transmitting zones are recommended. 

The source should be a small point of high intensity. This can 
be a Poirit-o-lite lamp or an illuminated pinhole about 1 mm. in 
diameter. If sunlight is available, the experiment may be 
performed in parallel light. Obtain an axial image with each 
lens, tilt the lens, and refocus. In general it will be difficult to 
observe the comatic images without tilting the lens to so large an 
angle that astigmatism will also be present. 

Make as accurately as possible a drawing of the comatic 
images with each lens. For which, if any, of the lenses or mir- 
rors used, is coma absent? 

Curvature of the Field and Distortion. For simple lenses these 
aberrations are marked with objects having large area. It is 
difficult, however, to distinguish between curvature of the field 
and astigmatism. The distortion due to a doublet of two convex 
lenses will be examined. Select two equal double-convex lenses 
of about 6 cm. aperture and 15 cm. focal length, each mounted 
in a tube so that the two may be placed different distances 
apart by sliding the lens tubes into another one which fits them 
tightly. For the source use a blackened photographic plate 
ruled with a rectangular grid of scratches about 1 cm. apart. 
This is to be backed with a sheet of thin white paper and illumi- 
nated from behind with a strong source of light. 

Measure the change in focus for regions of the image field at 
different distances from the axis. This will give the curvature 
of the field. Measure also the difference in magnification for 
different zones. This will give the distortion. Plot both the 
curvature and the distortion separately as ordinates on a single 
sheet of graph paper, against the distance from the axis as 
abscissa. 

Chromatic Aberration. Mount the plano-convex lens used 
in the determination of spherical aberration with a concentrated 
filament for a source. With a red filter, measure the focal dis- 
tance. Repeat with a blue filter. The difference in focus is a 
measure of the chromatic aberration. Repeat the experiment 
with an ordinary achromatic doublet. 



352 LIGHT: PRINCIPLES AND EXPERIMENTS 

EXPERIMENT 4 

MEASUREMENT OF INDEX OF REFRACTION BY MEANS 

OF A MICROSCOPE 

Apparatus. This consists mainly of a microscope mounted 
horizontally on a carriage in fairly accurate ways so that it is 
capable of horizontal motion of a few centimeters. The dis- 
tance moved is measured by means of a linear scale and a microm- 
eter head graduated to thousandths of a millimeter. In case the 
movable carriage and micrometer are not available, a tenth- 
millimeter scale may be attached to the horizontal microscope 
and the distance it is moved may be measured by means of a 
second, vertical, microscope focused on the scale and equipped 
with a micrometer eyepiece. In some cases high-grade micro- 
scopes are equipped with micrometers capable of measuring the 
focusing distance with great accuracy, so that no auxiliary 
measuring microscope is needed. 

There should also be an adjustable stand on which specimens 
and cells may be mounted in front of the horizontal microscope, 
and a source of light, preferably diffuse. 

Part A. Refractive Index of a Glass Block. The block to be 
examined should be one for which the index can also be measured 

by means of the grazing incidence method (see 
Experiment 7) and should therefore be a rec- 
tangular block with one end and two sides 
polished. The distance between the sides A 
and B (Fig. 1) is measured with a micrometer 
caliper to hundredths of a millimeter. Sprinkle 
a few grains of lycopodium powder on the 
surfaces. The microscope is first focused on a 
grain at A, and its position read, then on a grain at J3, and a reading 
taken. Since the angles of incidence of the rays entering the 
microscope are very small, to a high degree of approximation the 
distance the microscope is moved between readings is equal to 
the actual distance AB divided by the index of refraction of the 
glass. From this the index may be calculated. 

Part B. Refractive Index of a Liquid. Use one of the liquids 
to be examined in Experiment 7. First, on the inside walls of a 
dry cell with parallel glass walls like that shown in Fig. 2, sprinkle 
a little lycopodium powder and measure the distance between 




EXP. 5] 



THE PRISM SPECTROMETER 



353 



the walls by the method described in Part A. In this case, 

however, the distance moved by the microscope between readings 

will be the actual separation of the inner 

walls of the cell. Then remove the 

powder from the inner walls and sprinkle 

a little on the outer walls, and repeat the 

measurements. Then, without moving 

the cell, place the liquid in it, and re- 

measure the distance (apparent) between 

the outer walls. If a is the apparent 

distance measured between the outer 

walls without the liquid, b the apparent distance between them 

with the liquid, and r the actual distance between the inner 

walls, then 




Fio. 1. 



r - (a - 6) 

Answer the following questions : 

1. How many figures after the decimal place in the value of n are you 
justified in retaining? Explain your answer. 

2. What error in Part A would be introduced by having an angle of 
5 deg. between the normal to the plane surfaces and the axis of the 
microscope? 

3. What error in Part H would be introduced under the same 
conditions? 

4. Justify the assumption involved in the statement that the angle 
can be substituted for its sine in making the measurements in this 
experiment. 

EXPERIMENT 5 
THE PRISM SPECTROMETER 

Apparatus. A spectrometer equipped with a Gauss eye- 
piece, a prism of approximately 60 deg. refracting angle, an 
extra slit to be fitted over the telescope objective, a white light 
source, a mercury arc, a helium source. 

The adjustment of the spectrometer should be made first 
according to the directions in Appendix IV. It is recommended 
that the student read through these directions with a view to 
understanding their purpose, rather than follow them line by 
line without appreciating the significance of each operation. 



354 



LIGHT: PRINCIPLES AND EXPERIMENTS 




After the spectrometer is in adjustment, read the following 
experimental directions and make sure that each can be carried 
out without modification of the spectrometer. 

Part A. To Measure the Refracting Angle of the Prism. A 
prism is a fragile piece of equipment, easily chipped on its corners 

and edges so that its usefulness is impaired. It 
may be protected by mounting as shown in 
Fig. 1. If the base is somewhat larger than 
shown and equipped with three leveling screws, 
it will be easier to use. Adjust the prism table 
so that the faces of the prism are parallel to the 
vertical axis of the spectrometer. (It is essen- 
tial for this experiment only that the faces be 
perpendicular to the telescope axis, but it is 
assumed that the latter has been adjusted with 
reference to the axis of the spectrometer.) 

Method 1. Set the telescope perpendicular 
to one of the prism faces, using the Gauss 
eyepiece (see Sec. 7-3), and record the angle on 
the graduated circle. Repeat for the other 
prism face. The difference between the two readings, subtracted 
from 180 deg., gives the refracting angle A of the prism. 

CAUTION: in making this and all other readings it is essential 
that both verniers (or reading microscopes) be used. It is 
assumed that the telescope axis intersects the principal axis of 
the spectrometer, but this may not be exactly so. Any slight 
error in this respect may be eliminated by reading both verniers. 
This is illustrated by the following numerical example. 



FIG. 1. A 
mount to protect 
the prism, con- 
sisting of two cir- 
cular 
held 

gether by a rec- 
tangular plate 
against which the 
base of the prism 
is set. 





First 
Vernier 


Second 
Vernier 


First prism face 


276 25' 15" 


96 27' 32" 


Second prism face 


35 25 45 


215 29 10 








180 - A 


119 0' 30" 


119 I' 38" 


Mean 


119 


1' 4" 


A 


60 5 


8' 56" 









Each setting in this and other observations should be made 
four or five times and the mean value taken. 



EXP. 5] 



THE PRISM SPECTROMETER 



355 




Fu;. 2. 



Method 2. Project the light through the slit and collimator 
lens (white light will do) and set the prism on the table with the 
refracting edge at the center as in Fig. 2. Set the telescope in 
position I so that the reflected image of the slit in good focus is 
exactly at the intersection of the 
cross hairs. Repeat for position II. 
The angle between the two settings 
of the telescope is twice the prism 
angle. 

Part B. To Find the Index of 
Refraction and Dispersion Curve of 
the Prism by the Use of the Angle 
of Minimum Deviation. For this 
part of the experiment a discharge 
tube containing helium and mercury 
in correct quantities to yield spectra 
of approximately equal strengths 
is excellent, although two separate 

sources may be used. If the helium source is not available, some 
other such as hydrogen or neon may be used, or even sunlight. 
All that is required is a dozen or so easily distinguishable lines 
throughout the range of the visible spectrum. Do not use a slit 
that is too wide. The spectrum lines should be so narrow that 
they appear as sharp lines with no perceptible evidence of the 
width of the slit. A table of wave-lengths of suitable elements 
is given in Appendix VI. 

With the prism removed, and a mercury source in front of the 
slit, set the telescope and collimator in line so that the image 
of the slit falls exactly on the cross hairs, and record the'angle 
of the telescope. Then place the prism on the table as in Fig. 3. 
Do not move the collimator, but set the prism so that the light 
from the slit completely fills one face of the prism. Swing the 
telescope to one side and look directly at the spectrum in the 
other face of the prism. Rotate the prism table and prism 
about the vertical axis, first in one direction, then in the other, 
watching the spectrum at the same time. It will be noted that 
as the prism is rotated, the spectrum changes position, but for 
a particular angle of the prism the direction of motion of the 
spectrum reverses. That is, if the prism and collimator occupy 
the position shown in Fig. 3, as the prism is turned clockwise, 



35(5 LIGHT: PRINCIPLES AND EXPERIMENTS 

the spectrum will first seem to move to the right and then, revers- 
ing its motion, move toward the left, while the prism continues 
to rotate in the same sense as before. Locate the position of 
minimum deviation roughly in this way, swing the telescope into 
the field of view, and repeat the rotation of the prism back and 
forth over a smaller range until the position of minimum deviation 
for the green mercury line (wave-length 5.461 X 10~ 6 cm.) is 
accurately determined. The angle between this setting of the 
telescope and that recorded without the prism will be A, the 
angle of minimum deviation for 5461 angstroms. Calculate 
the index of refraction for this wave-length by means of eq. 8-5, 
page 89. Next obtain A for each of several bright lines through- 




Fio. 3. 

out the range of the visible spectrum, and calculate the corre- 
sponding indices of refraction. Plot the values of the index 
thus obtained against the wave-lengths, and obtain the dispersion 
curve. It is customary for manufacturers to specify for a 
particular glass the value of the index for the sodium doublet, 
whose average wave-length is 5893 angstroms. Obtain the 
value of the index of refraction for this wave-length from your 
curve and compare it with the value provided by the instructor. 
The dispersive power of a glass between the Fraunhofer C and G 
lines is given by the equation 



no /,>. 

W = - r- (1) 



Using the wave-lengths of the Fraunhofer C, D, and G lines 
(from Sec. 6-14), find n c , n/>, and no from your dispersion curve, 
and calculate the value of w. This result should also be com- 
pared with one provided by the instructor. 



Kxi>. 5] THE PRISM SPECTROMETER 357 

Calculate also the values of the Cauchy constants no and B 
from the following equations : 

Xi 2 X 2 2 (n 2 - 



Part C. The Resolving Power of the Prism. In order to 
apply Rayleigh's criterion it would be necessary to select two 
spectrum lines which are sufficiently close together so that they 
are just seen as distinct lines in the spectrometer under the best 
conditions. However, resolving power is shown by eq. 8-14 to be 
proportional to the equivalent thickness of the prism and thus 
to the width of the beam of light falling upon it. Hence we may 
choose any suitable pair of lines, such as the yellow pair in the 
mercury spectrum, 5770 and 5790 angstroms, which are much 
farther apart than those which are just resolved by the full prism, 
and decrease the width of the beam of parallel light passing from 
the collimator through the prism and thence to the telescope. 
This can be done by placing over the collimator or telescope lens 
in a vertical position a second slit whose width may be varied. 
This slit is then to be closed until the lines, each of which will 
appear to widen, are just on the point of becoming indistinguish- 
able. Carefully remove the second slit and measure its width a' 
with a microscope or comparator. The resolving power of that 
part of the prism thus used is given by 

R' = R, (3) 

where a is the width of the beam of light intercepted by the full 
prism face and R is the resolving power of the prism. But from 
geometry it is evident that w, the width of the face upon which 
the light is incident is given by 

t 
w = =--.- 



2 sin (A/2) 
and also, at minimum deviation, 

a = w cos i' = w cos i; 
hence, 



R'w cos i ,., 



358 



LIGHT: PRINCIPLES AND EXPERIMENTS 



Now we can calculate R' for the width a', from the fundamental 
definition of resolving power, i.e., 



Thus, by eq. 3, R can be obtained from the experimental results. 
For the average wave-length used, calculate also the value of R 
for the entire prism from 

2Bt 



K = 



X 3 



and compare it with the experimental result. Compute from 
R = aD where D is the dispersion of the prism, the values of the 
resolving power for several wave-lengths. 

EXPERIMENT 6 
THE SPECTROPHOTOMETER 

Spectrophotometers which make possible the comparison of 
the intensities of sources over small ranges of wave-length may be 
constructed in various ways. Two forms will be described here, 

in both of which the actual comparison 
is effected by the use of polarized light. 
The Glan Spectrophotometer. This 
instrument is similar to the ordinary 
spectrometer, but has certain modifi- 
cations making possible the comparison 
of two sources. The light from one of 
these passes directly into the upper 
portion Si of the slit (Fig. 1), while 
that from the other is directed into the lower portion $2 by 
means of a total reflecting prism. Inside the collimator tube, 
at a suitable distance from the slit, is placed a Wollaston prism 
(see Sec. 13-10). This prism divides the light from each of the 
sources into two beams which arc polarized so that the directions 
of their plane vibrations are perpendicular to each other. These 
four beams are refracted by an ordinary 60-deg. prism set on the 
spectrometer table, so that four parallel spectra are formed, one 
above the other. The middle two of these which we may call 
spectrum 2 and spectrum 3 are from /Si and Sz or vice versa, and 
consist of light vibrations which are in perpendicular planes. 
Spectra 1 and 4 do not appear in the field of view of the eyepiece. 
Between the collimator lens and the refracting prism is placed a 




FIG. 1. 




EXP. 6j THE SPECTROPHOTOMETER 359 

polarizing prism of the Glan type, in which the faces are inclined 

so that the transmitted beam undergoes no sidewise displacement 

as the prism is rotated. When this polarizing prism is oriented so 

as to transmit the full intensity of 

light reaching it from source 1, it 

will transmit none of the light from 

source 2, and when rotated through 

90 deg. from this position, it will A! 

transmit all the light reaching it 

from source 2 and none from source 

~i~ T M> .._. 

1. At intermediate angles, some of Jf 

the light from each source will reach 

Kic 2 

the eye, and for one particular angle 

of the polarizing prism the intensities of the two spectra will be 

equal for a given wave-length region. It is this particular angle 

which must be determined experimentally for each part of the 

spectrum. 

Referring to Fig. 2, and considering only the two middle 
spectra, let us suppose that the coordinate axis X represents the 
direction of vibration of the plane-polarized beam from source L, 
transmitted by the Wollaston prism, while Y represents the 
direction of vibration of the beam so transmitted from source 2. 
Let OH and 0V represent respectively the amplitudes of these 
vibrations. Then the ratio of these amplitudes is given by 
tan 6 = 0V /OH. If the Glan polarizing prism is oriented so 
that its plane of transmission is in the direction ON, making an 
angle 6 with OF, the component OA of the amplitude 0V will 
represent that part of the light from source 2 which will reach 
the eye. Similarly only the light represented by the amplitude 
component OA of OH will reach the eye, so that for the spectral 
region under observation the intensities of the two spectra 
will be the same. This will also be true if the polarizing prism is 
oriented so that its plane of transmission is parallel to ON'. 
Since the intensity is proportional to the square of the amplitude, 
it follows that when the Glan polarizer is set so that the two 
spectral regions have the same intensity, the ratio of the intensi- 
ties of the two original sources is given by 

Intensity 2 , . ,,,. 

.jr ,-^r = tan 2 0, (1) 

Intensity 1 



360 LIGHT: PRINCIPLES AND EXPERIMENTS 

where is the angle between the setting of the polarizing prism 
for equal intensities and the setting for complete extinction of 
the light from source 1. 

An ordinary spectrometer may be converted into a Glan 
spectrophotometer with a few changes. Across the middle of 
the slit is placed a strip which blocks the light and effectively 
divides the slit into an upper and a lower portion. The Wollaston 
prism is inserted in the tube so that the images from Si and $2 
(Fig. 1), coincide vertically. It should be possible to rotate the 
Wollaston through a small angle about the optical axis, and to 
move it along the collimator tube a short distance in either direc- 
tion. A nicol in a holder having a circle preferably graduated 
in four quadrants is fitted on the end of the collimator tube 
between the lens and the refracting prism. The eyepiece is 
equipped with a pair of long slit jaws which may be opened wide 
enough so as to permit a view of the entire spectrum, or closed 
so as to permit the transmission of a band of only about 25 to 30 
angstroms. This slit, or diaphragm, should be mounted at the 
focal plane of the eyepiece. 

Part A. The Comparison of Continuous Spectra. The dis- 
tribution of intensities of a number of sources may be compared 
with that of a standard lamp. While an accurate standard may 
be used, it is neither necessary nor advisable for ordinary class 
studies. It is satisfactory to use instead an ordinary new 40-watt 
Mazda lamp as an arbitrary standard. To this a sticker should 
be affixed near the base to indicate the side of the lamp from 
which the light is to be taken throughout a series of measure- 
ments, since the brightness of such a lamp is not the same when 
viewed from different directions. With this improvised standard 
several sources may be compared, as, for instance, an old- 
fashioned carbon filament lamp, a ruby lamp, a gasoline Welsbach 
lamp, or sunlight. The intensity distribution in the light from a 
second 40-watt, 110-volt lamp operated on 135 to 150 volts may 
also be studied, if voltages higher than 110 are available, or a 
lamp of lower rated voltage than 110 may similarly be overrun. 

To place the two spectra exactly one above the other, a mer- 
cury arc or other bright-line source may be used in front of the 
slit with the total reflection prism removed. If the bright lines 
due to light passing through Si and S 2 are not exactly in line, 
the Wollaston prism should be rotated slightly until the spectra 



EXP. 6] THE SPECTROPHOTOMETEli 361 

coincide vertically. At the same time, if the two .spectra are not 
exactly edge to edge, a slight displacement of the Wollaston 
prism may be made along the tube to bring them together with- 
out overlapping or separation. 

The prism which refracts the light is next to be calibrated. 
Setting it for minimum deviation approximately for the yellow, 
using any convenient lines, such as the sodium lines or the 
mercury lines, the angle of the telescope should be plotted against 
wave-length for a number of positions in the spectrum. Either 
sunlight or ordinary laboratory sources will serve for this purpose. 
It is not necessary to read the angle of the telescope closer than 
Y deg. for any setting, since the comparisons to be made later 
are of regions of the spectrum several angstroms in width. 
Enough readings should be taken, however, so that a graph 
may be made, with wave-lengths as abscissas and telescope 
settings as ordinates. After the points are plotted, a smooth 
curve should be drawn through them, so that for any subsequent 
position of the telescope the corresponding wave-length may be 
quickly read from the graph. 

Next set up the standard lamp and the source to be compared 
with it. If clear glass lamps are used, a piece of finely ground 
glass should be placed between each lamp and the slit, and as 
close to the former as possible without the risk of breakage by 
heat. Glass ground to a sufficiently fine grain may be made by 
grinding lantern-slide cover glasses with fine carborundum, or 
emery, and water, using a flat piece of iron as a tool. When the 
lamps are accurately in place and the zero position of the polarizing 
prism determined, set the telescope on a region in the red end of 
the spectrum, rotate the polarizing prism until the spectra 
of the two sources appear the same intensity, and record the 
setting of the polarizer. Make several settings for each position 
in the spectrum, and average their values. Then move the 
telescope to successive regions of the spectrum about 100 or 
200 angstroms apart, and make similar observations. Calculate 
the intensity ratios by means of eq. 1, and plot the values 
obtained. (Remember that the telescope gives an inverted 
image of the slit. 

Part B. Absorption of Colored Transmitting Substances. 
Using the same adjustments as in Part A, allow the light from the 
standard source to pass through both parts of the slit, and place 



362 LIGHT: PRINCIPLES AND EXPERIMENTS 

a piece of didymium glass about 5 mm. thick over the lower part. 
Measure and plot the relative transmissions for each spectral 
region as outlined above. Other substances whose absorption 
may be measured are solutions of potassium permanganate, 
cobalt chloride, and thin films of metal evaporated or otherwise 
deposited on glass. A very useful chart of the transmissions of 
about 50 substances is to be found on page 16 of Wood's "Phys- 
ical Optics," 1911 edition. Extensive tables of the spectral 
transmissions of substances are also to be found in the "Hand- 
book of Chemistry and Physics," published by the Chemical 
Rubber Company, and others are procurable from the Corning 
Glass Works, the Eastman Kodak Company, and Jena Glass 
Works (Fish-Schurman Company, agents). 

Part C. (Optional) Relative Intensities of Bright Lines in a 
Spectrum. By using a slit of sufficient width so that each bright 
line from a source such as the mercury arc appears as a narrow 
rectangle of light, the intensities of lines in discontinuous spectra 
may be compared with the spectrum of a white-light source. 
This comparison, however, cannot be used to obtain the relative 
intensities within the discontinuous spectrum itself unless the 
distribution of intensity in the white-light source is known. 
While this distribution can be measured for any continuous 
spectrum, it is perhaps best to use a standardized and calibrated 
white-light source for this part of the experiment. 

A tungsten ribbon lamp operated at a sufficiently high tempera- 
ture may be said to radiate in accordance with the Wien distri- 
bution law, which may be written 

E\ -V C2/XT 

\5 > 

where E\ is the energy radiated for a particular wave-length X, 
T is the absolute temperature, and Ci and c% are constants. The 
value of c 2 is commonly taken as 1.433 cm.-deg. On p. 363 is a 
calibration for brightness temperature of a special standard 
lamp, made of a tungsten ribbon filament. 

This lamp, operated on a current of 26 to 28 amp., may be 
said to approximate a black body. Hence, by Wien's law, the 
energy distribution in its spectrum may be calculated. If this 
distribution is known, it is possible to obtain with a fair degree 
of accuracy the distribution of intensity in the visual part of a 



EXP. 6] 



THE SPECTROPHOTOMETER 



363 



spectrum which is compared with it. However, the student is 
to be warned against undertaking this part of the experiment 

RELATION BETWEEN CURRENT AND TEMPERATURE 

Current, amp. Brightness temperature, K. 

13.0 1,484 

15.5 1,667 

18.5 1,856 

22.5 2,076 

27.0 2,300 

32.5 2,549 

without careful supervision by the instructor, since calibrated 
lamps are expensive and may easily be ruined. 

In reporting this experiment, answer the following question: 
In adjusting the Wollaston prism so that the two spectra aro 
exactly edge to edge, it will probably be found that when they 
are so adjusted in the red, they will overlap in the blue or vice 
versa. Explain why this should be so. 




FIG. 3. The Brace-Lemon spectrophotometer. C\ and Cz, collimators; N\ and 

N, nicols; T, telescope; B, Brace prism. 

The Brace -Lemon Spectrophotometer. Another type of spec- 
trophotometer which may be used with considerable precision is 
the Brace-Lemon, illustrated in Fig. 3. It has two identical 
collimators, in one of which are set two Glan polarizing prisms. 
The prism nearer the collimator lens is fixed in azimuth, while 
the other may be rotated. The dispersive instrument is a Brace 
compound prism made of two 30-deg. prisms. On one of these 
is deposited an opaque coat of suitable metal of high reflecting 



364 LIGHT: PRINCIPLES AND EXPERIMENTS 

power, which covers the middle section of the prism face, the 
dividing edges between this coat and the unobstructed portions 
of the face being perpendicular to the refracting edge of the 
prism. Sometimes the reflecting coat is deposited on one half 
of the prism face, the upper or lower, and the remaining half 
left transparent. The two 30-deg. prisms are then sealed 
together with Canada balsam, so that the reflecting coat is 
between the two halves of the compound prism. 

This instrument has the advantage that when the compound 
prism is properly made and adjusted with reference to the 
optical path of the light, the dividing line between the two 
spectra to be compared is rendered invisible. 

In order to make a comparison of two sources, one of them is 
set before each slit, and the rotatable prism, whose zero position 
has been previously determined, is turned until the intensities 
in a given spectral region are the same. 

Since the Brace compound prism is fragile and the adjustments 
must be made very exactly, it is perhaps desirable that the 
instrument be of a fixed form, and not adjustable by the student. 
In case adjustments are necessary, they may be made as follows: 

1. The spectrometer should be adjusted with the Brace prism removed 
in the same manner as for any similar instrument, the essential require- 
ments being that the axis of each collimator and that of the telescope 
should be perpendicular to the vertical axis of the instrument, and that 
the axis of rotation of the telescope should coincide with that vertical 
axis. 

2. The rotatable polarizing prism A'i should be set for minimum 
transmission; i.e., its plane of transmission should be set parallel to 
that of iV 2 . 

3. The Brace prism should be replaced, leveled so that its faces are 
parallel to the vertical axis, and set at the proper height. It should 
then be set for minimum deviation for the yellow. 

4. The prism should be calibrated in the manner outlined for the 
Glan spectrophotometer, and a calibration curve constructed by means 
of which the approximate wave-length region be found for any angle of 
the telescope. 

6. The collimator slits should be the same width. For this purpose, 
it is essential that each should be provided with a micrometer head. 
It will be necessary to determine the zero, i.e., the setting for which no 
light passes through the slit, for each micrometer, since the slit rnay not 
be closed when the head reads zero. 



Kxr. 7] WDEX OF REFRACTION BY TOTAL REFLECTION 365 



6. The two spectra must then be made to coincide vertically. This 
may be done with the aid of a mercury arc from which the light is 
reflected into both collimator tubes. Two arcs may be used, one in 
front of each collimator. If the spectral lines from one are displaced 
sideways with respect to those from the other, unclamp the collimator 
tube containing the Glan prisms, arid rotate it slightly to eliminate the 
displacement. Be sure to reel am p the tube. 

The measurements may be made in the same manner as out- 
lined for the Gian speetrophotometer. 

EXPERIMENT 7 
INDEX OF REFRACTION BY TOTAL REFLECTION 

There are several refractometers, i.e., instruments for the 
measurement of refractive index, which make use of the principle 
of total reflection. Perhaps the best known is that of Abbe, 
which gives excellent results, especially for liquids. But since 
the use of a commercial Abbe refractometer wovdd give the 
student very little practice in optical manipulation, 
an experimental arrangement using the same 
principles is employed, in which the means of 
measuring the angles is an ordinary spectrometer. 

Before doing the experiment, Sec. 8-7 should be 
read carefully. 

Apparatus. A pair of flint-glass prisms of high 
index of refraction, which may profitably be 
mounted as shown in Fig. 1, a rectangular block 
of glass with all sides polished, a number of liquids 
whose indices are to be measured, a spectrometer, 
a monochromatic source, and an ordinary con- 
densing lens of about 2 in. aperture and 6 or 8 in. 
focal length. Handle the glass pieces carefully as they are fragile 
and difficult to replace. 

Part A. The Index of a Glass Prism. The spectrometer must 
be in adjustment as in Experiment 5, with the telescope focused 
for paralleL light. See that the prisms are thoroughly clean 
and place one of them on the spectrometer table. If it is neces- 
sary to clean them, it can be done either with alcohol or ordinary 
commercial acetone. The use of good soap and lukewarm water 
is also recommended. Using a Gauss eyepiece, measure the 
refracting angle of the prism. Move the collimator to one side, 




FIG. 1. 



366 LIGHT: PRINCIPLES AND EXPERIMENTS 

or, if it is rigidly mounted, rotate the prism table. Adjust the 
source and condensing lens so that a broad beam illuminates the 
entire diagonal face of the prism. Looking into the face AB 
(Fig. 2) at the illumination, with the telescope swung to one 
side, rotate the prism table until the field of view observed at e 
is divided by a sharp vertical line on one side of which the field 
is not illuminated. It is essential that no light from the source 
fall on the side BC, which can be covered with a screen of card- 
board. The bright side of the field will obviously coincide with 
light which is incident upon the diagonal face at angles less than 
90 deg., and in the case illustrated in Fig. 2 it will be on the left 
of the field of view. As the table and prism are rotated, the 




bright field will become narrower and will be seen to have a 
sharp vertical edge. A precaution to be taken at this point is 
to be sure that this apparent division between light and dark is 
not an image of the boundary of the source itself. This can be 
done by shifting the source or condenser lens slightly from side to 
side, observing carefully meanwhile to see whether the apparent 
division also moves. The telescope is now to be swung into the 
field of view, and the cross hairs set accurately on the line of 
division. Record the setting of the telescope on the divided 
circle. Then by means of the Gauss eyepiece set the telescope 
perpendicular to the face AB of the prism and record the angle. 
The difference between these two settings will be the angle i' 
of eq. 8-19. The index of refraction may now be found by 
means of eq. 8-20 since the medium m in this case is air, the 
index of which is practically unity. Putting n m 1, and 
solving for n g , we have. 



- ( sin i' + cos 4V 2 

fig ty A "l 1. 

sin 2 A 

Since the angle i' may be either on the left- or right-hand side 
of the normal to the surface AB, it may be either positive or 
negative; in the illustration in Fig. 2 it is positive. 



KXP. 7] INDEX OF REFRACTION BY TOTAL REFLECTION 367 



Part B. Index of Refraction of a Liquid. Liquids which may 
be conveniently tested are: Distilled water, glycerin, and cedar 
oil which has been prepared for oil immersion in microscopes. 
Put a drop of the liquid whose index is to be measured on tho 
diagonal face of one of the prisms, and lay the other gently on it, 
so that the two form a rectangular block. Do not press them 

B 





FIG. 3. 

together tightly. Set them on the prism table and illuminate as 
shown in Fig. 3. Rotate the table until the division of the field 
into light and dark areas is in the field of view, as in Part A. 
The selection of the dividing boundary will be aided in this 
case by the fact that interference fringes are formed in the thin 
film of liquid between the prisms. These fringes will in general 
appear tangent to the boundary, as in Fig. 4. If the prisms are 
pressed together too tightly, the fringes will be so sharp and 





FIG. 4. The boundary 
is the vertical division in 
the middle. 



FIG. 5. 



conspicuous as to make the setting of the cross hairs on the 
boundary difficult. As in Part A, record the setting of the 
telescope for the boundary and the normal to the surface. 
Remember that i' may be either negative or positive. By means 
of eq. 8-19 the index of refraction of the liquid may be calculated. 
Part C. Index of Refraction of a Glass Plate. Thoroughly 
clean the 45 deg. prisms. Put on the surface of one of them a drop 
or two of some liquid whose index is higher than that of the 
glass plate, and place the block gently on the face AC, as shown 
in Fig. 5. Liquids suitable for the purpose are: Methylene 



368 LIGHT: PRINCIPLES AND EXPERIMENTS 

iodide, index 1.74; a-monobromnapthalene, index 1.66; aniline, 
index 1.56. Locate the boundary between the light and dark 
fields as before. It will be distinguished as in Part B by the 
presence of interference fringes. There may be two boundaries, 
one corresponding to the index of the glass block, the other 
to the index of the liquid. The one to be chosen is that corre- 
sponding to the smaller index. Carry out the measurements 
as in Part B, and calculate the index of refraction for the glass 
block by means of eq. 8-20. 

EXPERIMENT 8 

WAVE-LENGTH DETERMINATION BY MEANS OF 

FRESNEL'S BIPRISM 

The theory of the Fresnel apparatus is to be found in Sees. 10-6 
and 10-7. 

Apparatus. An optical bench about 2 m. long, a mercury 
arc, a filter transmitting only 5461 angstroms, an accurate slit, a 
biprism mounted in a rotatable holder provided with a circular 
rack so that the biprism may be rotated about an axis perpen- 

Filkr. >-S/it Cross- 



FIG. 1.--- Arrangement of apparatus for use of Fresnel oiprism. 

dicular to the common prism base, a high power micrometer 
eyepiece with cross hairs, an achromatic lens of about 30 cm. 
focal length and 5 cm. aperture. On account of the dimness 
of the fringes, it is advisable to perform this experiment in a 
separate room, or in a space sufficiently screened so that there 
is no disturbance from other light sources. 

Part A. The apparatus is shown in Fig. 1. Be careful to place 
the biprism not more than about 35 cm. from the slit. Make 
sure that the cone of light from the slit covers the biprism, and 
have the slit quite narrow. Adjust all parts so that the optical 
axis is horizontal arid centered with respect to each part. Set 
the common face F of the biprism toward the slit and perpen- 
dicular to the direction of light. This may be done with sufficient 
accuracy by sighting down the biprism and, if need be, getting 
the image of the slit which is reflected from F into coincidence 
with the slit opening. Move the eyepiece along the bench out 



EXP. 8] FRESNEL'S B1PRJSM 369 

of the way and place the eye at E in the figure. A very fine 
pattern of fringes will be seen between the two virtual images of 
the slit. Next place the eyepiece, previously accurately focused 
on the cross hairs, at E and look through it. Be sure of at least 
four settings. Let Si, s 2 , s 3 , s 4 , and s 5 be those fringe positions 
on the left-, and si', s 2 ', s 3 ', /, and s 6 ' be those on the right-hand 
side of the pattern. Suppose also that Si is the 30th fringe 
from 81. Then the value of e obtained from 



Si - Si 



is 30 times as accurate as that obtained by measuring the distance 
between any two consecutive fringes, since any lack of precision 
in making the settings is divided by 30. Thus five independent 
determinations of e may be made, each of a high degree of 
accuracy, from which a final average value of the spacing of the 
fringes may be obtained. 

To measure d, place the lens between the biprism and the 
eyepiece. Provided the distance between these is more than 
four times the focal length of the lens, there will be two positions 
of the lens for which real images of the slit will be in focus at the 
plane of the cross hairs. With the eyepiece micrometer, measure 
the separations d\ of the virtual images for the first position, and 
r/ 2 for the second position. The value of d is given by 

d = 

By means of equation 10-9 calculate X. 

Hand in the answers to the following in your report : 

a. Derive the equation d = -\/di X ^2- 

6. Enumerate the possible sources of error in your result. 

c. Calculate the probable error of your final result for X. 

Part B. (Optional.) The Use of Parallel Light. If a lens 
is placed between the slit and the biprism with its principal 
focus at the slit, parallel light will be incident upon the latter. 
The position of the lens should be determined with great care. 
This can be done by using a telescope, previously adjusted for 
parallel light, in place of the eyepiece, with the biprism removed. 
With the telescope in place, move the lens until a sharp image 
of the slit is in good focus at the cross hairs. Replace the biprism 



370 LIGHT: PRINCIPLES AND EXPERIMENTS 

and micrometer eyepiece, and measure e as in Part A. Since the 
wave-length with parallel light may be calculated by means of 
eq. 10-11, the only remaining measurement is that of the angle 25, 
between the virtual images. This may be obtained with a 
telescope which can be turned about a vertical axis and whose 
angle of rotation can be measured with great accuracy. The 
angle of rotation may be measured by means of a mirror mounted 
on the telescope so that the deviation of a beam of light may be 
found, or the telescope may be equipped with a micrometer and 
scale. 

Part B may also be done on a spectrometer, in which case the 
biprism is mounted at the center of the spectrometer table 
and the angles measured on the divided circle. 

Part C. (Optional.) Measurement of 3. with Fresnel Mir- 
rors. In case a biprism is not available, a pair of mirrors may be 
used, inclined at a very small angle. In this case the observations 
are the same as in Part A for the biprism, except that instead 
of d, the angle a between the mirrors is to be found and X calcu- 
lated by means of eq. 10-5. 

EXPERIMENT 9 

MEASUREMENT OF DISTANCE WITH THE 
MICHELSON INTERFEROMETER 

The theory of this instrument is presented in Sees. 11-4, 5, 6, 
and 7. 

Apparatus. A Michelson interferometer, equipped with three 
plane mirrors A, B, and B' as shown in Fig. 1, a mercury arc, a 
filter transmitting only the green mercury line 5461 angstroms, 
a white-light source. 

Adjustment of the Interferometer. The mirrors A, B, and B' 
should have good reflecting surfaces. If they are made of glass, 
metallic coatings may be deposited by one of the methods 
outlined in Appendix V. The dividing plate C, which should 
be cut from the same plane parallel plate as the compensator D, 
may have on its front surface (nearer the observer) a semi- 
transparent metallic coat which will transmit half the light to 
B and B' and reflect half to A. This mirror should not be 
handled unnecessarily as the half coat is fragile and easily rubbed 
off. Fringes may be obtained without the half coat, and even 



EXP. 9] THE MICHELSON INTERFEROMETER 371 

with surfaces of poor reflecting power at A, B, and B', but the 
beginner will find it much easier to work with good mirrors. 

After the source and condenser lens are set as indicated in 
Fig. 1 for good illumination of the mirrors, the fringes may bo 
found. From theory it is seen that the position of formation 
of the fringes depends upon the relative distances of A and B 
from C. If these distances are the same, not only will the fringes 
be formed at a position corresponding to good reading distance 
from the eye, but for monochromatic light as well as for white 
light the visibility will be near maximum. The first step, there- 
fore, is to move the carriage supporting A by means of the main 
screw until AC and A B are about equal. A match stick may 
then be mounted by means of laboratory wax vertically at M. 

A 




The image of the end of the stick should appear in the center 
of the field of B. Cover B' with a card. There will be seen 
four images of the stick. Two of these are due to light divided 
at the metal-coated surface at C and two are due to light divided 
at the other surface of the plate C. The first two are easily dis- 
tinguished since they are more distinct. By means of the 
adjusting screws behind B, bring them into coincidence, where- 
upon the fringes should appear. Some manipulation may be 
required before the fringes are actually seen, since they will 
probably be very narrow at first. Once they are seen, move the 
adjusting screws behind B carefully until the fringes are of 
suitable width. About 20 in a field 2 cm. wide is usually a 
satisfactory spacing. If the fringes are not straight, it means 
either that the surfaces are not plane, or that the distances AC 
and BC are not equal. Ruling out the first possibility, the 
fringes may be made straight by turning the main screw which 
moves the carriage of A in either direction. The screw should 
be turned until the curvature of the fringes in one direction is 



372 LIGHT: PRINCIPLES AND EXPERIMENTS 

pronounced and then reversed until the curvature is in the other 
direction. The position sought can then be found between 
narrow limits. If the fringes change their slope while this is done, 
the track on which the carriage moves is not clean, and should be 
wiped with a clean cloth and a little oil. B' may now be adjusted 
in the same manner, and a distance in front or behind B of 
about 0.1 mm. 

Different purposes may be served in this experiment, some of 
which require more elaborate adjustments than are possible 
with the instrument as described. It should be kept in mind, 
however, that the procedure is satisfactory, no matter how 
simple, if it teaches the student how to manipulate the inter- 
ferometer and understand its basic principles. The purpose of 
this experiment, therefore, is to assist in the understanding of 
the optical phenomena. To demonstrate the accuracy obtain- 
able in comparison with that of a micrometer microscope, the 
distance between the mirrors B and B' will be measured. 

Part A. Comparison of the Measurement of Distance by 
Fringe Counting and by a Micrometer Microscope. For this 

it will be necessary to have a micro- 
meter microscope rigidly attached to 
the bed of the interferometer and 
focused on a fine line or ruling at right 
angles to the motion of A, and on a 
plate attached to the carriage of A. A tenth-millimeter scale 
ruled on glass may be used. 

If B and B' are parallel, the virtual sources with which we 
may replace them are indicated in Fig. 2 by the parallel lines 
b and 6'. The dotted line a\ is the virtual source which replaces 
A when white-light fringes are obtained in A and B. When 
white fringes are obtained in the common field of A and B', the 
mirror will have been moved through the distance d to a position 
indicated by the virtual source a*. This distance can be meas- 
ured roughly, (1) by means of the micrometer head on the main 
screw which moves A, (2) by the micrometer microscope, and 
(3) by counting the number of monochromatic fringes which 
pass a point in the field of view. Since the total path distance 
introduced by method (3) between B and B' is 2d, we have 
2d nX, where n is the number of fringes which pass a given 
point in the field. 




EXP. 9j THE MICHELSON INTERFEROMETER 373 

It is necessary to locate the positions of A for which white-light 
fringes are obtained in turn in B and B'. This can be done by the 
following steps: 

1. Adjust B and B' until the fringes in both are straight, of equal 
width, and parallel. 

2. Using a white-light source held between the mercury arc and the 
interferometer, move A back and forth slightly until the white-light 
fringes appear in either B or B'. (There may be some lost motion in the 
bearings of the main screw, but the angle of turn required to take this 
up may be easily found by observing the monochromatic fringes.) 
If white-light fringes are found in B first, and B' is farther away from (' 
than B, next move A to correspond to the position of B' and find the 
white-light fringes. It should then be possible to move A from one 
position to the other two or three times so as to get the corresponding 
angle of turn on the micrometer head of the main screw. 

3. With the white-light fringes in view in either B or B', take up the 
lost motion in the direction which will bring them into the other field. 

It is now necessary to establish a fiducial point for counting 
fringes. This can be done elaborately in a variety of ways, but 
a satisfactory arrangement is a pair of wires, one fixed at M 
and one at N (Fig. 1). If the head is kept in a position so that 
these two are superposed, there should be no mistake in counting 
fringes because of a shift of the point of view. (For persons of 
normal vision, or with good accommodation, there will be no 
difficulty in seeing the fringes and the wires clearly; for persons 
who are farsighted, it may be necessary to mount between the 
eye and the interferometer a lens of 15 to 25 cm. focal length.) 

Set the moving cross hair of the micrometer microscope on 
the ruled line or scratch on the carriage at A, repeating the 
setting several times in the same direction, and recording each 
observation. 

Count the monochromatic fringes which pass as the main 
screw is turned slowly until the black central fringe seen in B' 
is coincident with the fiducial mark. For slow turning of the 
screw there is usually provided an auxiliary worm gear which 
can be brought to bear against the edge of the micrometer head. 
If the distance between B and B' is 0.1 mm., the number of 
fringes of the green mercury line (5.461 X 10~ 6 cm.) which pass 
should be about 367. 



374 LIGHT: PRINCIPLES AND EXPERIMENTS 

Again set the moving cross hair of the micrometer microscope 
on the ruled line, which will have been moved, repeating the 
setting several times as before. The distance through which the 
ruled line and hence the mirror A has been moved should be 

n\ 



where n is the number of fringes counted. 

Repeat the count, moving A in the opposite direction. 
Also repeat the settings on the ruled line with the micrometer 
microscope. 

Answer the following questions : 

1. Compare the accuracy of your observations by the two methods. 

2. In making the preliminary adjustments with the mercury arc, did 
you notice some fringes, elliptical in shape, superposed on the fringes to 
be measured? If so, attempt an explanation of their origin. 

3. Remove the compensating plate D, and look for the white-light 
fringes. To what is their appearance due? What do you conclude 
regarding the function of the compensating plate? 

Part A'. The Use of Circular Fringes. If circular fringes 
are used, it will not be necessary to use any fiducial mark, since 
these fringes are formed when the mirrors are parallel. Hence, 
no matter where the eye is placed in the field of view, the fringes 
will suffer no changes in diameter. Hence there is no difference 
of phase introduced by an accidental shifting of the head. In 
this case, it is necessary to count the fringes which appear from 
or disappear into the center of the system as one of the mirrors 
is moved. It is a little harder to count circular fringes, how- 
ever, unless the light is strictly monochromatic, especially when 
the path difference AC BC is small. If the path difference is 
small the circular fringes are very large, and for zero path 
difference they cannot be distinguished at all. Under these 
circumstances it will be impossible to keep accurate count. 

Part B. The Calibration of a Scale. If mirrors B and B' 
are mounted on a carriage which may be moved in accurate 
ways by a screw in the same fashion as is mirror A, the distance 
BB' may be used to step off some other distance which is several 
times BB' . This may be done by moving A from coincidence 
with B to coincidence with B' } then moving B and B' until B is 



EXP. 9] THE MICHELSON INTERFEROMETER 375 

once more in coincidence with A, and so on until the full distance 
in the larger step is measured. If this larger distance is not an 
even multiple of BB', some difficulty is experienced, unless the 
remainder can be measured by an independent method. Sup- 
pose, for instance, the larger distance is the spacing between two 
scale marks on a standard millimeter scale. Then 

K = n(BB') + a fraction of BB', 

where K is the larger distance, n is the whole number of times 
BB' is stepped off. The fraction may be measured in the 
manner already described by a count of monochromatic fringes. 

Part B'. The Calibration of a Scale. Alternative Method. 
In case the interferometer is one with only a single mirror at B, a 
tenth-millimeter scale may be calibrated with a not excessively 
large fringe count. A scale ruled on glass with divisions of 0.1 
mm. is recommended. This can be attached to the carriage of A, 
and extended beyond the bed of the interferometer so that it 
may be illuminated from beneath, with a vertical microscope 
equipped with a micrometer eyepiece directly above the scale. 
It is recommended that a point of light or small source be used, 
placed so as to give, in effect, dark-field illumination. In other 
words, instead of placing the source directly on the optical axis 
of the microscope, place it so that the light diffracted by the 
scale divisions is seen in the microscope. 

White-light fringes are not needed. Instead, straight fringes 
of monochromatic light may be used. The procedure is to set 
the moving cross hair of the micrometer microscope on the 
image of one of the scale divisions S\. Several settings should 
be made and the average of the micrometer readings taken. 
Then mirror A is moved until the micrometer cross hair is over 
another scale division Sz, the fringes being counted meanwhile. 
It is important to realize, however, that bringing the cross hair 
of the micrometer microscope into coincidence with Si and $2 
in the ways described above are not comparable procedures. 
The setting on Si is the average of a number, rapidly made, while 
that on Sz can be made only once, since fringes are being counted, 
and is then made slowly. Consequently the distance from Si 
to Sz measured in this way may be in error. This possibility 
of error can be checked by setting the cross hair once more on S* 
by turning the micrometer head, making several such settings 



376 LIGHT: PRINCIPLES AND EXPERIMENTS 

in the identical manner in which the cross hair was formerly set 
on Si. Any difference is to be added or subtracted from the 
rated scale distance Sz Si. For instance, suppose that when 
the micrometer microscope was originally set on Si the average 
of the readings on the head was 87.5, the pitch of the micrometer 
screw being 1 mm. That when the corrective setting was 
finally made on Sz, the average of the readings on the head 
turned out to be 88.8. The difference in the settings is 0.013 mm. 
This amount is to be added to or subtracted from the rated dis- 
tance between Sz and Si, depending on whether increased readings 
on the head correspond to an advance of the cross hair toward 
$2 or a motion back toward Si. 

Another method which may be successful if it is practiced 
before the experiment is under way is as follows: Start counting 
fringes a little before the cross hair reaches Si, and when the 
observer at the microscope decides the cross hair is just on *Sj, 
note the extent of the count. Also, continue counting a little 
after the cross hair is seen to reach S%. If HI is the number in 
the count as the cross hair passes Si, and w 2 is the number in the 
count as it passes $2, then n 2 n\ is the number of fringes 
corresponding to Sz S\. 

(Sz -Si) = (rh HI) = 

Obviously, if the rated distance Sz Si is known accurately, 
the wave-length of the light may be found, instead of the scale 
being calibrated. 

EXPERIMENT 10 

MEASUREMENT OF INDEX OF REFRACTION WITH 
A MICHELSON INTERFEROMETER 

Theory. If a plane-parallel plate of index of refraction n is 
inserted normal to the path of one of the beams of light traversing 
the arms of a Michelson interferometer, the increase of optical 
path introduced will be 2(n l)t, where t is the thickness of the 
plate. The factor 2 occurs because the light traverses the plate 
twice. For monochromatic light of wave-length X, the difference 
of path introduced is N\, where N is the number of fringes dis- 
placement introduced when the plate is inserted. Hence, if a 



EXP. 10] INDEX OF REFRACTION BY INTERFEROMETER 377 

Michelson interferometer is adjusted for white-light fringes, a 
parallel plate of index n inserted in one of the paths, and a count 
made of the number of fringes which cross the field when equality 
of optical path is reestablished, it would be possible to measure 
n with a high degree of accuracy. This is not a satisfactory 
method of measuring the index of refraction, first, because N 
is too large a number to be conveniently counted unless the 
plate is very thin; second, because it is extremely difficult to 
determine the center of a white-light-fringe pattern when the 
two arms of the interferometer contain unequal thicknesses of 
glass. If, however, a parallel plate in one of the arms is rotated 
through a small measured angle, the path of the light will be 
changed, and the number of fringes N corresponding to this 
change may be counted. The 
exact method of performing this 
experiment will be described in 
a later paragraph. 

The change of path through 
the glass plate depends upon 
the thickness of the plate, the 
angle through which it is 
turned, and the index of re- 
fraction. The last of these 
three may be calculated if the 



o 



/v 


/ 




/ \ 


-J 




/ \ 






X 


4-a 




I 


^ 


1 


// 


x x 


A 

1 
1 


^ / ' 

x d 


\ 
6 ^^x 


* 

1 

1 

y 


rf-/-i'!t-N 




^ 
\ 


/ e ^r 

/ e 1 


'^jT - 





/ 



\ 



\ 



FlQ 



other two are measured. Let 
OP (Fig. 1) be the original 
direction of the light normal to plate of thickness t. The 
total optical path between a and c for tho light going in one 
direction is nt + be. After the plate is rotated through an angle 
i, this optical path has been increased to ad - n + de. Hence 
the total increase of optical path, since the light travels over the 
path twice, is 

2(ad - n + de - nt - be} = N\. (1) 

But 

A 

ad 



cos r 



de = dc sin i (jc fd) sin ? = t tan i sin i t tan r sin i, 



hr t 

\J\s ~ . (/ 

COS I 



378 LIGHT: PRINCIPLES AND EXPERIMENTS 

So, 

nt . . , ... , , . . . t N\ 

---- \- i t a n i sin i t tan r sin i nt ---- : -\- t = -^r-- 
cos r cos i 2 

Using Snell's law, n sin r = sin i, this may bo reduced to 

n(l - cos 02* - A r X = (2t - iVX)(l - cos + ^-- (2) 

Since the last term is small compared to the others, it may be 
neglected, leaving for the index of refraction 

(2t - N\)(l - cosQ 
27(T - cos - ' ^ } 



In the experiment, two such plates, PI and P%, are used, one in 
either arm of the interferometer. These are made only half as 
high as the mirrors A and B so as to permit the observation in 
the field of view above them of fringes unaffected by the change 
of angle i. The use of two plates insures equal optical paths 
in the two arms, at all times when the angles of these plates 
with the direction of the light beams are the same, making 
possible the observation of white-light fringes through the plates 
when they arc tilted at the same angle with the beam. 

First, by the method outlined in Experiment 8, obtain in the 
upper part of the field vertical white-light fringes. This had 
better be done with the half plates Pi and Pz already in place, 
as inserting them afterward may be the cause of an accidental 
displacement of the other parts of the interferometer. With 
the white light fringes obtained, next set Pi and PZ normal to 
the light path as nearly as can be done while looking down on the 
instrument. Then, while observing the fringes, turn PI slowly 
until the fringes appear also in the lower part of the field. Now 
observe what happens if half plate P 2 is rotated a slight amount 
in one direction. If the lower fringes move completely out of 
the field and do not return, rotate P 2 in the other direction. 
What will usually happen is that either in turning one way or the 
other the fringe system will be displaced a number of fringes, say, 
to the right, and then move in the opposite direction. This 
indicates that the half plate Pi was not, in the rough adjustment 
of the plate normal to the light path, set normal with sufficient 
precision. Hence it is to be rotated by such an amount that 



EXP. 10] INDEX OF REFRACTION BY INTERFEROMETER 379 

eventually the white-light fringes in the lower part of the field 
will move continuously out of the field in one direction upon a 
turn of Pz in one sense, and out of the field in the same direction 
with a turn of PZ in the opposite sense, without returning in 
either case. If the half plates PI and PI are cut from the same 
parallel plate, i.e., are of exactly the same thickness, the white- 
light fringes should coincide in the upper and lower parts of the 
field. 

Sometimes it is impossible to obtain the adjustment described 
in the preceding paragraph. This may be due to the fact that 
one of the half plates is "leaning" slightly in its frame, a con- 
dition which may be corrected by rocking the plate slightly. 
Another reason for lack of adjustment may be that the half 
plates are not cut from a parallel plate, but from one which has a 
slight wedge shape, the two sides being out of parallelism by a 
fringe or two. In this case the fringes in the upper and lower 
parts of the field of view of the interferometer may not be 
parallel, and one of the plates should be turned over in its frame. 

After the white-light fringes extend across both the upper and 
lower portions of the field, and the half plates are precisely 
normal to the beams, turn P\ through an angle of about 15 deg. 
This should be done in the direction in which the last adjustment 
of that plate was made, so that there is no lost motion to be 
taken up. (If no micrometer attachment is available for deter- 
mining exactly the angle that P\ is turned through, a small 
mirror fastened to the cell for Pi and facing in the direction of a 
telescope and scale placed about 6 ft. away may be used. The 
angle will then be measured in the conventional manner with the 
telescope and scale.) Having turned Pi, and measured its 
angle, slowly turn Qz through the same angle, meanwhile count- 
ing fringes to the number (N) of monochromatic light which 
pass a selected point in the field, until the white-light fringes 
reappear in the lower part of the field and coincide with those 
in the upper. For this purpose, the green line of mercury may 
be used, and a source of white light be held or clamped in such a 
way that part of the field is illuminated by it. Thus the mono- 
chromatic fringes may be observed to pass, and at the same time 
the white-light fringes will be detected when they appear. An 
excellent check on the value of N is then to turn P 2 in the opposite 
direction, meanwhile counting fringes, until the white-light 



380 LIGHT: PRINCIPLES AND EXPERIMENTS 

fringes once more appear in coincidence. P-t will then have been 
turned through twice the angle i, and the number of fringes in 
this second count should be 2N. 

Remove Pz and measure its thickness t with a micrometer 
caliper. Then calculate the value of n, using cq. 3. 

Answer the following questions : 

1. What percentage of error is introduced in the measurement of the 
index of refraction by an error of 10 min. of arc in the measurement of 
the angle through which Pa is turned from the normal position? 

2. What percentage of error is introduced in the measurement of the 
index by an error of 0.005 mm. in the thickness of /Y? 

3. What percentage of error is introduced in the measurement of the 
index by an error in the count of N of five fringes? 

4. Would any appreciable improvement in the result be obtained by 
retaining the last term in eq. 2? 

EXPKRIMP;NT 11 

THE RATIO OF TWO WAVE-LENGTHS WITH THE 
MICHELSON INTERFEROMETER 

Read Sec. 11-7 for the discussion of visibility fringes. 

Apparatus. A Michelson interferometer, a mercury arc, a 
filter of didymium glass about 5 mm. thick, aqueous solutions of 
copper nitrate, cobalt sulphate, and nickel acetate, an assortment 
of gelatin filters, a condensing lens. Uranine may be substi- 
tuted for the solution of cobalt suphato. 

Part A. The solutions are to be prepared of sufficient density 
so that the combined filter will permit the transmission of only 
X4358 and X5461. It is essential that these be of about the same 
visual intensity. The transmission of the filter can be tested 
with a direct-vision spectroscope or a spectrometer and 60-deg. 
prism. Of the stronger mercury lines, copper nitrate transmits 
only those from X4046 to X5790, inclusive. Nickel chloride 
cuts out X4046 and X4071. Didymium glass cuts out X5770 
and X5790, and cobalt sulphate or uranine cuts out the faint 
green line X4916. These solutions may be mixed together in a 
filter cell about 1 cm. thick, or better still, in separate cells. If 
there is any precipitate present, the addition of a little hydro- 
chloric acid will remove it. 

Adjust the interferometer for white-light fringes. With the 
mercury arc and the combined filter the succession of maxima 



EXP. 11] THE RATIO OF TWO WAVE-LENGTHS 381 

and minima will be clearly seen. Count the number of fringes 
from the minimum closest to the center of the white-light pattern 
to the thirtieth minimum away. Obtain by division an average 
number of fringes N between any two consecutive minima. N 
will not necessarily be a whole number. The ratio of the two 

N I 
wave-lengths will be given by ^ Since this ratio is so far 

from unity, N being small, it will be necessary to try both the 
positive and negative signs. 

Part B. Remove the didymium glass and substitute a filter 
which transmits only X5770 and X5790. This may be composed 
of an orange gelatin filter with an aqueous solution of cobalt 
sulphate. Since the two wave-lengths are almost the same, 
many fringes (between 200 and 300) must pass the field of view in 
passing from one position of maximum visibility to another. 
Moreover, at minimum visibility the counting of the fringes will 
be impossible or nearly so. Instead it will be necessary to rely 
on the accuracy of the main screw which moves the carriage. 

First calibrate the screw by turning the micrometer head 
through about one-tenth of a turn, counting the fringes mean- 
while. Then set for the first position of minimum visibility 
next to the position of zero path difference, and read the microm- 
eter. Turn the screw until the twentieth minimum comes into 
the field, keeping track meanwhile of the total number of turns 
of the screw. The fractional part of a turn may be read from the 
micrometer. The total number of fringes which have passed 
may thus be calculated. 

Since the screw may not possess an accuracy warranting 
this calculation, an alternative method is suggested as a check. 
Substitute a filter transmitting only the green line of mercury, 
and turn the screw through exactly one revolution, counting 
fringes meanwhile. This will determine the pitch of the screw. 
(If this is already accurately known, the foregoing will not 
be necessary.) Then, if, say, the screw wap turned through 
3.32 turns in passing from one minimum of visibility to a distant 
one for X5770 and X5790, since the distance moved is equal to 
ATX/2, we have 

2 X 3.32 X 0.05 = N 
0.0000578 



382 LIGHT: PRINCIPLES AND EXPERIMENTS 

where the pitch of the screw is taken as 0.05 cm. and the average 
wave-length is used. In the example, the number of fringes is 
about 5743, corresponding to the passage from a given mini- 
mum to the twenty first from it. Hence 5743/21, or 273.48 
fringes would pass in going from one minimum to the next, and 
273.48/274.48 is the ratio of the wave-lengths. For the accuracy 
possible it is not material whether in (N 1)/AT the -f- or 
sign is used. Assuming the correct value for the longer wave- 
length to be 5790.66, the value for the shorter becomes 5769.56. 
The correct value is 5769.60. 

EXPERIMENT 12 
THE FABRY-PEROT INTERFEROMETER 

In the discussion of the theory of this instrument in Sec. 8-11 
it has been pointed out that the use of multiple beams instead of 
double beams to produce interference results in a great decrease 
in the width of the interference maxima. Thus the observer is 
permitted to see, distinctly separated, the interference fringes 
due to two or more radiations. Since, however, for each mono- 
chromatic radiation, the interference pattern consists of a, set- 
of concentric rings, there is no direct method of finding out which 
of two wave-lengths may be the larger. For instance, in the 
interference pattern of the complex mercury line A = 5461 illus- 
trated in Fig. 11-17, it is not immediately possible to say whether 
the faint fringes shown are of shorter or longer wave-length 
than the brighter ones. If, however, however, two radiations 
already well known are used, the ratio of their wave-lengths 
may be found. 

Apparatus. A Fabry-Perot interferometer with one plate on a 
movable carriage so that it may be moved perpendicular to its 
face by means of a screw, a mercury arc, a condensing lens, and a 
filter transmitting only 5770 and 5790 angstroms. 

Since the sharpness of the fringes depends on the number of 
reflections between the two plates, care should be taken to see 
that the reflecting coats are as bright as possible. Aluminum 
deposited on the plates by evaporation is exceedingly durable. 
The coats of metal should be much thicker than half coats in 
order to insure high reflecting power. Care should be taken to 
see that the ways in which the carriage moves are clean and free 
from dust. 



EXP. 12] THE FABRY-PEROT INTERFEROMETER 383 

In finding the fringes, which are circular, it is best to have the 
separation between the mirrors as small as possible, since then 
the diameters of the innermost fringes are very large. 

Do not jam the mirrors together. 

Set up the mercury arc and condensing lens so that the entire 
area of the plates is well illuminated. Hold a pencil or match 
stick between the rear mirror and the lens, and manipulate the 
adjusting screws in front of the interferometer until the manifold 
images of the stick coincide. The fringes will then be seen, 
probably poorly defined and as if astigmatic. Careful adjust- 
ment is then made by turning the screws, which change the tilt 
of the fixed mirror, meanwhile observing whether the diameters 
change as the eye is moved from side to side and up and down. 
If on moving the eye to the left, the circles become larger, the 
distance between the mirrors on the left is greater than that on 
the right, and further adjustment should be made. In this 
manner, get the two mirrors as nearly parallel as possible. Then 
run the movable carriage back a few millimeters and see if the 
parallelism is lost. If, to any appreciable extent, it is, the ways 
must be cleaned again or other necessary steps taken to improve 
the mechanical performance. 

Next insert the filter transmitting only the two wave-lengths 
to be observed. A tentative turn or two of the main screw will 
show that, as the carriage moves, single and double fringes 
alternate. If the metallic coatings are not very thick, the 
resolving power will be less, and instead of definite doubling of 
the fringes, there will be simply a decrease in visibility of the 
interference pattern. 

The radius of any circular fringe increases as the mirrors are 
moved apart, and decreases as they are brought together. A 
bright fringe for a given wave-length \i has a radius which 
depends on the separation of the mirrors, the orders of inter- 
ference of the fringe, and the wave-length, as given by the 
equation 

2e cos i = PiXi, 

where e is the separation of the mirrors, Pi is the order of inter- 
ference, or number of wave-lengths difference of path between 
two interfering beams, and i is the angle subtended by the radius 
of the fringe. Hence, if there is another longer wave-length \2 



384 LIGHT: PRINCIPLES AND EXPERIMENTS 

whose fringes have the same radii as Xi, i.e., if the fringes for the 
two are exactly superposed, then 

2e cos i = P 2 \2. 

As the distance between the two mirrors is increased, in passing 
from one position of coincidence to the next, the change in NZ 
will be one less than the change in N\. Hence, if N is the number 
of fringes which appear at the center of the pattern as the distance 
is increased, then 

(N - 1)X, = #Xi. 

Actually, the wave-lengths are so nearly alike that it is not 
possible to tell which set of fringes belongs to the longer, and the 
procedure is to count the number of fringes which appear at the 
center between two successive coincidences (or maxima of 
visibility) and obtain the ratio of the wave-lengths by the equation 

X, N 1 
X 2 N 

If one of the wave-lengths is known, the other may be calcu- 
lated. Determination of the point at which the fringes are 
exactly superposed is difficult. However, the error in this 
determination can be reduced by counting from one maximum 
or point of superposition to the fifth or sixth from it, and obtain- 
ing a mean value for N. An alternative procedure is to count 
from the position of minimum visibility instead. If the fringes 
are very sharp, this will correspond to the position where the 
two sets are midway between each other, with dark rings of 
equal width and blackness between them. 

EXPERIMENT 13 

MEASUREMENT OF WAVE-LENGTH BY DIFFRACTION 

AT A SINGLE SLIT 

For the theory of diffraction by a single slit, see Sec. 12-8. 

It is evident from eq. 12-5 that the intensity obtained by 
diffraction of light through a single slit becomes zero for values 
of <p m/a, where a is the width of the slit and m is an integer. 
Since <p sin i sin 0, satisfactory experimental conditions 
will exist if i is made zero, so that sin 6 m/a. It must be 
remembered, however, that eq. 12-5 is based on the assumption 



EXP. 13] WAVE-LENGTH BY DIFFRACTION 385 

that the light illuminating the slit is collimated, so that the 
pattern is one obtained by Fraunhofer diffraction. 

Apparatus. A spectrometer, a mercury arc, a filter for the 
transmission of the green mercury line 5461 angstroms, an 
auxiliary slit. (In case a spectrometer is not available for this 
experiment, satisfactory results may be obtained if the primary 
slit, upon which the light of a mercury arc is focused through a 
filter, is placed at a distance of about 20 ft. from a laboratory 
telescope, which has an auxiliary slit fitted over its objective. 
The telescope must be capable of rotation about a vertical axis, 
and there must be provided also some method for measuring 
this rotation to an accuracy of about 5 sec. of arc. The use of a 
spectrometer is advised.) 

If a filter transmitting only the green line is not available or 
if its use dims the light too much, the light from the collimator 
may first be passed through a prism, and thereafter the green 
line allowed to fall upon the auxiliary slit. 

With a spectrometer correctly focused so that an image of the 
primary slit is at the plane of the cross hairs, and a satisfactory 
filter, or prismatic dispersion, set the auxiliary slit with its jaws 
at the center of the spectrometer table so that the plane of the 
jaws is perpendicular to the beam of light from the collimator. 
Adjust the width of the auxiliary slit so that the fringes are dis- 
tinct and of measurable width. Frequently there is difficulty in 
getting sufficient light intensity to permit accurate settings on the 
diffraction minima. In this case it is not good practice to open 
the primary slit at the source end of the collimator too wide, since 
this results in a blurring of the pattern. The primary slit 
should be closed to as small dimensions as will afford good 
visibility of the fringe pattern. Proper shielding of the instru- 
ment from extraneous light, and allowing time for the eye to 
become accustomed to conditions, will be of advantage. 

It is recommended that before observations are begun, the 
auxiliary slit be removed carefully and its width measured by 
means of a micrometer microscope or comparator. This may be 
done afterward, but in case there is any danger of altering the 
slit width by moving it, time will be saved by finding it out at the 
start. 

The quantities to be measured are the width a of the auxiliary 
slit and the angle between two successive minima. Since the 



386 



LIGHT: PRINCIPLES AND EXPERIMENTS 



angle to be measured is very small, sin 6 may be put equal to 6, so 
that = w/a, from which 6 may be calculated. The easiest 
settings to make are obviously those on minima which are close 
to the middle of the pattern where the intensity is greatest. 
However, increased accuracy may result if as many minima as 
possible are measured, and a weighted mean of the resulting 
value of be found. If the light is not too faint, it should be 
possible to set on eight or ten minima on either side. In case 
difficulty is experienced in seeing the cross hairs when they are 
set on minima, flashing a utility light of low brightness into the 
telescope will help. 

As in other experiments, there should be several settings made 
for each position of the telescope, and the mean taken in each 
case. It is possible to obtain a final average value of B by simply 
subtracting the mean value of each setting from the mean value 
of the next contiguous one, thus obtaining several values of 6 
which may be averaged. It should be pointed out, however, that 
the result thus obtained is not the mean of independent observa- 
tions. Furthermore, the observations are not of equal weight. 
A better practice is to subtract the setting on the first minimum 
on one side from the first minimum on the other side of the 
central image, yielding a value twice 0, and assign to it a weight 
of 2; then subtract the setting on the second minimum on the 
one side from the setting on the second minimum on the other 
side, yielding a value four times 0, with a weight of 4, and so on, 
as far out as minima can be distinguished. This practice of 
weighting, however, which assumes that all the observations are 
of equal difficulty, should be modified in the present case since 
the minima become progressively more difficult to distinguish 
as one goes further out from the center of the pattern. The 
following modification is suggested. Suppose the settings on the 
two seventh minima are as follows : 



Side 


Angle of telescope 
(degrees omitted) 


Deviation 
from mean 


Left: 






(1) 


42' 15" 


9" 


(2) 


41' 8" 


58" 


(3) 


42' 17" 


9" 


(4) 


42' 34" 


28" 




Mean 42' 6" 


23.5" 



EXP. 14] 



THE DOUBLE-SLIT INTERFEROMETER 



387 



Side 


Angle of telescope 
degrees omitted 


Deviation 
from mean 


Right: 






(1) 


17' 2" 


21" 


(2) 


16' 27" 


14" 


(3) 


16' 38" 


3" 


(4) 


16' 37" 


4" 




Mean 16' 41" 


10.5" 



Suppose, similarly, the settings on the two sixth minima have 
values of the mean deviation of 18.2 and +7.6 min. Then 
obviously the value of 6 determined from the seventh minima 
is not l %2 times as accurate as that determined from the sixth, 
and should not be weighted as much in arriving at a mean. If 
the results obtained by the student are such that he is in doubt 
as to the proper procedure, he should consult the instructor 
before arriving at a final determination of 9. 

Having found a and 6, calculate the value of the wave-length 
of the light used. 

Repeat the experiment for a different value of a. 

Answer the following questions : 

1. What error in the wave-length is caused by an uncertainty of 
0.005 mm. in the width of a? What percentage of error? 

2. What error in the wave-length is caused by an uncertainty of 
25 sec. of arc in the value of 0? What percentage of error? 

3. What error is due to both these uncertainties combined? What 
percentage of error? 

EXPERIMENT 14 
THE DOUBLE-SLIT INTERFEROMETER 

The theory involved in this experiment is to be found in 
Sec. 12-9, on Diffraction through Two Equal Slits. 

Apparatus. A good laboratory telescope with an objective 
of about 25 to 35 cm. focal length, provided with a high-power 
eyepiece. In front of the objective is to be mounted a specially 
constructed double slit, each opening of which can be adjusted 
in width over a range of about 3 mm. This double slit must 
have bilateral motion, so that the slits may be separated to any 
distance between about 6 and 30 mm. The telescope should 
preferably be mounted in a snug fitting tube, so that it may be 



388 LIGHT: PRINCIPLES AND EXPERIMENTS 

rotated about its axis, in order to adjust the double slits to a 
vertical position. If this is not feasible, the double slit should 
be rotatable. A single slit to be used as a source, accurately 
round pinholes, filters, and a mercury arc are also needed. A 
rotatable biprism is necessary for Part E, in case that part is done. 

Adjustments. Set the single slit in a vertical position, and 
illuminate it with a mercury arc, provided with a filter to permit 
the passage of the green line X5461. An image of the arc must 
be projected on the slit, so that the latter is truly a source with 
respect to the observing telescope. The single slit and the double 
slits must be vertical, or at least parallel to each other. 

It will be necessary to adjust the width of the single slit so that 
the resulting interference pattern can be made to disappear (or 
reach a minimum visibility) within the range of motion of the 
double slit. Obviously, from the equation 

l\ 
w= d' 

in which w is the slit width, and I the distance between the source 
slit and the telescope, a rough preliminary calculation of the 
most desirable value of w will save a great deal of time. For a 
value of I of about 8 m. and a separation d of the double slits of 
about 1.5 cm., w would be about 0.3 mm. 

The double slits should be equal in width. A suitable width, 
for other dimensions given, is about 1 mm., although a smaller 
width can be used if there is sufficient light intensity. A greater 
width will, of course, result in a more brilliant image, but the 
diffraction pattern from each slit will be narrower, and the 
number of fringes in the brightest portion of the image will be 
less. When the telescope is correctly focused, the image seen 
in the eyepiece will be similar to the two-slit diffraction pattern 
seen in Fig. 12-146, except that there will usually be more and 
narrower interference fringes. As will be seen from the theory, 
the number and spacing of the interference fringes will depend 
upon the ratio of the common width of the double slits to the 
width of the opaque space between them. 

Part A. With proper illumination on the slit, and the maxi- 
mum intensity of the cone of light directed to the telescope, 
adjust the source slit to a width between 0.2 and 0.3 mm. Focus 
the telescope on the slit and mount the double slit in front of the 



EXP. 14] THE DOUBLE-SLIT INTERFEROMETER 389 

objective, so that the source slit and the double slits are parallel. 
Fringes should be seen. Beginning with the widest separation, 
slowly reduce the separation of the double slits until the fringes 
disappear. Record the distance from center to center of the 
double slits at this point, and continue narrowing the separation 
until the last disappearance is observed. It is sometimes 
difficult, especially for the beginner, to detect the point of 
disappearance, because (a) the slit source may be slightly 
wedged-shaped, in which case disappearance will not occur 
simultaneously along the fringes; (6) the fringes are so narrow 
that they are indistinct; (c) the two slits are not of the same 
width, and only a minimum visibility is attained. In case of 
failure to observe disappearance or to identify it as a first-order 
disappearance, the theoretical separation d of the double slits may 
be approximately calculated by eq. 1, to aid in the observations. 

White light may be used instead of the mercury arc, since the 
order of interference is quite small. In this case, in the calcula- 
tion of w from eq. 1, the value of X to which the eye has maximum 
sensitivity should be used. For most eyes this is between 
5.5 X 10~ 5 and 5.7 X 10~ 5 cm. Make several determinations 
of d and I and calculate w and the mean error of observation. 

Measure the width of the single slit carefully with a microm- 
eter microscope, a traveling microscope, or a comparator, repeat- 
ing the measurements for different places on the slit. Replace 
the single slit in its former position in the optical train. 

Part B. Set the double slit at a separation of about 1 to 2 cm. 
and vary the width of the single slit slowly, observing the succes- 
sive widths for which the fringes disappear. For this purpose 
it is desirable that the slit be equipped with a micrometer head 
for quick reading. If this is not available, the width for each 
disappearance must be measured as before. The widths observed 
should be multiples of some value which, within experimental 
error, will be the value of w obtained in Part A. The allowable 
error of the experimental determination of w by observation 
of the disappearance of the fringes in Parts A and B is between 
2 per cent and 8 per cent, depending upon the care and skill of 
the operator and the precision of the mechanical parts. 

Part C. Remove the double slit and the eyepiece and substi- 
tute a micrometer eyepiece. Can you measure the width of the 
slit? If so, with what accuracy? What limits the accuracy in 



390 LIGHT : PRINCIPLES AND EXPERIMENTS 

this case? Compare the accuracy of this measurement with 
the ones made in Parts A and B. 

Part D. Remove the single slit and substitute a pinhole 
in a thin metal sheet. Use white light for greater intensity. In 
this case the angular diameter is given by 1.22\l/d, when the 
fringes disappear. 

Part E. Place the rotable biprism over the pinhole so that 
the two images seen through the biprism just touch edges when 
viewed through the telescope without the double slit. Replace 
the double slit, orient it to the angle of the biprism, and measure 
the separation of the two virtual images by the disappearance 
of the fringes as before. The distance between the centers of 
the two virtual images of the pinhole is given by a = 0.61XZ/*/. 
Repeat Part C for this source. 

EXPERIMENT 15 

THE DIFFRACTION GRATING 

The theory of this experiment is to be found in Sec. 12-12. 

Apparatus. A spectrometer, a mercury arc, a helium dis- 
charge tube, a plane diffraction grating of the reflection type, a 
Gauss eyepiece. If only a transmission grating is available, the 
following procedure must be modified slightly. 

Adjust the spectrometer as directed in Appendix IV. Place 
the mercury arc in front of the slit, which may be opened to a 
width of about 0.5 mm. By looking at the face of the collimator 
lens, make sure that the entire lens is filled with light. Set the 
grating on the spectrometer table so that (a) its plane contains 
the main vertical axis of the spectrometer, (6) the cone of light 
from the collimator is centered on the ruled area of the grating, 
(c) the rulings are parallel to the axis of the spectrometer, (d) 
the slit is parallel to the rulings. 

Adjustments for (a) and (6) may be initially made by simple 
inspection, with the telescope swung out of the way. To insure 
that the axis actually lies in the surface, the Gauss eyepiece 
method may be used. Assuming that the telescope is in adjust- 
ment so that its axis cuts, and is perpendicular to, the axis of the 
spectrometer, the latter will be parallel to the grating surface 
when the image of the cross hairs reflected from the surface of 
the grating coincides with the cross hairs themselves. 



EXP. 15] THE DIFFRACTION GRATING 391 

Before making adjustment (c), fasten a wire or match across 
the middle of the slit, and set the spectrometer table so that the 
angle of incidence is between 50 and 60 deg. Then each line 
of the spectrum, in as many orders as can be reached on both 
sides of the central image (direct reflection), should be examined 
through the telescope. When the middle of the slit stays the 
same height in the field of view, the rulings are parallel to the 
axis. 

Before attempting adjustment (d), ascertain whether the slit 
is in a fixed position, or if it may be rotated. If the former is the 
case, do not risk damaging it by forcing rotation. If rotatable 
about the axis of the collimator, turn it until the image of the 
obstacle is sharpest; then the slit will be parallel to the 
rulings. 

At this point it is well to caution the student that precise 
adjustments and observations are not possible unless the telescope 
is properly focused on the spectrum lines. Remember also 
that even if the instrument is fitted with so-called achromats, the 
focal lengths will not be the same for all wave-lengths. It is also 
well to find out if the grating has Rowland or Lyman ghosts. l If 
these exist, they should be ignored in making the observations. 

Before continuing, narrow the slit until the lines are as sharp 
as possible. This stage will be reached when closing the slit 
further makes no apparent change in the width of a line but 
simply a reduction in the intensity. 

Part A. With the foregoing adjustments made, next find the 
setting in angle on the circle of the spectrometer, (a) when the 
telescope is normal to the grating, (6) when the central image 
(direct reflection) is on the cross hairs, (c) when each strong 
line of the mercury spectrum, in each order which can be reached, 
is on the cross hairs. Tabulate these data. In the first column 
put the order of the spectrum; in the second, the wave-lengths; 
in the third, the angles. From settings (a), (6), and (c) the 
values of i and 6 may be found and put in columns four and 
five. The grating equation is 

1 Rowland ghosts are false images of a spectrum line, grouped symmetric- 
ally on both sides of the line; they are usually faint in low orders. Lyman 
ghosts are false orders of spectra occurring at angles for which m is not a 
whole number (see eq. 1). Both types are due to irregularities in the ruling 
of the grating. 



392 LIGHT: PRINCIPLES AND EXPERIMENTS 

' i /> fH\ , ., x 

sin z + sin 6 = T-> (1) 

in which i is the angle of incidence, the angle of diffraction, m the 
order, X the wave-length, and d the grating space. From eq. 1 
and the observed data, the wave-lengths of the lines may be 
calculated if the grating space is known, or vice versa. The nega- 
tive sign is used when the spectrum is on the same side of the 
normal to the grating as the central image. 

Part B. Resolving Power. Since ft = X/dX when the limit 
of resolution is reached, the resolving power is found by deter- 
mining the limit of resolution which, according to Rayleigh's 
criterion, is reached when two spectrum lines are a distance apart 
such that the central maximum of the diffraction (or interference) 
pattern of one line falls upon the first minimum of the other. 
With the particular grating used, it would be difficult to find two 
spectrum lines which just fulfill this requirement. Moreover, it 
is essential that the entire grating be uniformly illuminated. 
The procedure is, then, to find the resolving power r of a small 
width w of the grating, and calculate R for the entire grating 
from the relation 

if _ entire width W of grating 

width w illuminated to obtain r ' 

In order to use only a small portion of the grating, place over 
the telescope lens an auxiliary slit. When this is closed to a 
width a, the value of w is given by w = a/cos 0. 

At this stage the student should make sure that he has a 
satisfactory source of a close pair of lines. For moderately low 
resolving power the yellow doublet of mercury at 5770 and 
5790 angstroms is suitable, but if the grating has rulings as 
close as, say, 5000 to the centimeter, its resolving power will be 
so high that the auxiliary slit width must be very small. For 
the ordinary reflection grating, therefore, the sodium doublet 
at 5890 and 5896 angstroms is 'most satisfactory. There are 
many sources of this light, perhaps the most brilliant being that 
obtained when an oxygas flame is trained on a small piece of 
pyrex tubing held on an iron rod about ^ in. in diameter. If 
oxygen is not available, a well-adjusted air-gas flame and soft 
glass tubing will do. 



EXP. 15] THE DIFFRACTION GRATING 393 

With the proper source in operation in front of the first slit, 
close down the auxiliary slit until the two lines of the doublet can 
just be observed separately. Make several trials. Carefully 
remove the auxiliary slit and measure its width a with a microm- 
eter microscope. Find and calculate w. 

The resolving power of a grating is given by 

It = mn, (3) 



where m is the order, and n is the total number of rulings. From 
the known value of the grating space and the measured value 
of w calculate r. Then find R from eq. 2. Compare this 
with the value of R given by the definition of resolving power: 
R = X/dX (when the limit of resolution is reached). 

Part C. Dispersion. This is defined as D = dQ/d\. Calcu- 
late Z> from the values of 6 for two close lines such as the mercury 
yellow lines, and compare it with D obtained from 

D= m 



d cos 6 

Discuss the errors and their probable origin, in your determina- 
tions of the wave-lengths. 

Part D. The Transmission Grating. The foregoing direc- 
tions are for a reflection grating of fairly high dispersion. In 
case only a transmission replica is available, the directions will 
apply with slight modifications. Instead of the angle of the 
direct reflection, that of the transmitted beam, i.e., at 180 deg. 
from the collimator setting, must be used. Also, in eq. 1 the 
positive sign is used if the diffracted light is on the same side 
of the normal as the incident beam. Other modifications may be 
suggested by the instructor. 

Part E. The Concave Grating. In case only a concave grat- 
ing is available, essentially the same quantities may be found 
by experiment, but the procedure will be quite different, depend- 
ing on the type of mounting used. There are three general 
ways of mounting a concave grating: 

a. The Rowland Circle. If light is incident at an angle i 
with the normal to the grating, then the position of the astigmatic 
spectral line is given by the equation for the primary astigmatic 
focus (eq. 6-8), with 6, the angle of diffraction, substituted for 



394 



LIGHT: PRINCIPLES AND EXPERIMENTS 



i', and n and ri put equal to unity. Solved for the distance s 1 
to the spectral line, this equation becomes 



ps cos' 2 



s(cos -f- cos i) p cos 2 i 



(4) 



where p is used instead of r for the radius of curvature of the 
grating. If s = p cos i, then s f = p cos 0, and the grating, the 
slit, and the spectral line lie on a circle called the Rowland circle 
shown in Fig. 1. Mountings in which the slit, grating, and part 
or all of the Rowland circle are arranged in a fixed position are 



Grating 




* Source 
FIG. 1.- -The Rowland circle. 




Slit _ 

holder 

FIG. 2. -The Rowland 
mounting. The straight lines 
intersecting at the slit are the 
tracks A and B. 



called Paschen mountings. To measure wave-lengths with this 
mounting it is necessary to know accurately the angles i and 6 
for each line. The method by which these may be found can be 
worked out by the instructor. 

b. The Rowland Mounting. From the geometry of the circle 
it follows that two lines which intersect at right angles at the 
slit will cross the circle, one at the center of the grating, the other 
at a point in the focal plane for which in eq. 4 is zero, as shown 
in Fig. 2. When the grating and eyepiece (or photographic 
plateholder) are mounted on the extremities of a bar of length p, 
this may be slid along two tracks A and B (Fig. 2) and different 
parts of the spectrum observed. This is called the Rowland 
mounting and has the advantage that for small angles 0' on 
either side of the normal, the dispersion is uniform. From eq. 1 
and the constants of the instrument the wave-length of a line 



EXP. 16] SIMPLE POLARIZATION EXPERIMENTS 395 

exactly on the normal may be found, after determining i and 8. 
This may be repeated for each line, or else the dispersion may be 
found for a small region near the normal, and an interpolation 
method used. 

c. The Eagle Mounting. Sometimes it is convenient to set 
the grating so as to utilize the diffracted light which is returned 
directly back along the same path as the incident beam, or 
nearly so, as shown in Fig. 13-35 for a prism instrument. This 
type of mounting is little used except for photographic spectro- 
scopy. In case it is necessary to use it for visual determinations 
of wave-length, it is recommended that Baly, "Spectroscopy," 
Vol. I, be consulted. 

EXPERIMENT 16 

SIMPLE POLARIZATION EXPERIMENTS 

The theory of this experiment is to be found in Chap. XIII. 

Apparatus. A polariscope 1 of design similar to that supplied by 
the Gaertner Scientific Co., illustrated in Fig. 1, a white light 
with diffusing bulb, a monochromatic source such as a mercury 
arc and filter for X5461, or a sodium burner (which need not be 
very bright), a supply of the polarizing materials mentioned in 
the directions given below. 

Procedure. 1. Make a dot on a piece of white paper and over 
it place a rhomb of calcite 1 cm. thick or more. Rotate the 
rhomb and identify the o- and e-rays. Which travels the faster 
through the crystal? This may be determined by seeing which 
image appears to be nearer the upper face. The one which 
travels the slower should be nearer. Why? Does this identify 
calcite as a positive or negative crystal? Check your conclusions 
with the theory in Chap. XIII. 

2. Tilt the rhomb so that the light is transmitted in a direc- 
tion nearly parallel to the optic axis. If a crystal is at hand 
which is flattened and polished in two planes cut perpendicular 

'Less expensive devices are sold in which the analyzer and polarizer 
consist of the patented substance "polaroid." These are extremely useful, 
but cannot be used for so wide a variety of experiments as the older polari- 
scopes unless they are completely equipped and have polaroid of the best 
quality sealed between good optical glass plates. These experimental 
directions are written for a complete polariscope, and may be modified by the 
instructor if only a simpler device is available. 



396 



LIGHT: PRINCIPLES AND EXPERIMENTS 



to the optic axis, so much the better. What do you observe 
regarding the apparent positions of the two images when viewed 
in this direction? 

3. Arrange the frosted-bulb white-light source and a screen 
with an opening o as shown in Fig. 2. The instrument may be 
vertical or at any convenient angle. 




Flo. 1. The polariscope. (Courtesy of Uaertner Scientific Co./ 

4. Adjust mirror m (Fig. 2) at such an angle with the axis of 
the instrument that the light through the opening o is incident 
at about 57 deg. Mirror m is then the polarizer. At the upper 

s end set the blackened mirror m' which 
will then act as the analyzer. Set m' so 
that the beams om, mm 1 , and m'e (Fig. 2) 
are in the same plane. Rotate m' about 
a horizontal axis so that the angle of 
incidence of mm' is about 57 deg. Then 
turn m' about an axis parallel to mm' , 
watching the reflected light meanwhile. 
The light should be extinguished when 
m'e makes an angle of 90 deg. with om. 

If extinction is not complete, a slight adjustment of tho angles 

of m and m' may make it so. 

5. Remove m' and substitute the pile of plates at an angle of 
57 deg. with mm'. Look through these from above and rotate 



ol. 




BXP. 16] SIMPLE POLARIZATION EXPERIMENTS 397 

about an axis parallel to mm 1 as before. At which angle is the 
extinction most complete? This illustrates the almost com- 
plete polarization by successive transmission through successive 
surfaces, each one of which is at the polarizing angle. 

6. Remove the pile of plates and substitute the Nicol prism 
as an analyzer. Rotate this about axis mm' until the light is 
extinguished. Then place the rhomb of calcite on a rotable 
holder in the path mm' . Rotate the calcite until the ordinary 
ray is extinguished. Then rotate the analyzer until the o-ray 
reappears and the e-ray is extinguished. Do the results of these 
observations confirm the statement that the plane of vibration 
of the o-ray is perpendicular to the principal section of the caicite 
and that of the e-ray is parallel to it? Explain in detail. 

7. Replace the white-light source with a monochromatic 
source. Remove the rhomb of calcite, turn the analyzer to 
extinction, and replace the rhomb with a half-wave plate for the 
wave-length of the source. What do you observe when the 
half-wave plate is rotated? Set the plate at the position for 
extinction, record the angle, and turn it in either direction 
through about 25 deg. Then note the setting in angle of the 
nicol, turn the nicol until the light is extinguished once more, and 
note the angle through which the nicol has been turned. This 
angle should be twice 25 deg., or 50 deg. From the fact that the 
analyzer can extinguish plane-polarized light incident upon and 
transmitted by a half-wave plate, what do you infer regarding 
the nature of the light vibration so transmitted? 

8. Remove the half-wave plate, set the analyzer for extinc- 
tion, and replace the half-wave plate with a quarter- wave plate, 
set at any angle at random. Now rotate the analyzer. Can 
you extinguish the light? If so, turn the quarter-wave plate 
through an angle of about 15 deg. and try again. 

Remove the quarter-wave plate, set the nicol for extinction, 
replace the quarter-wave plate set for extinction, and note the 
direction of its principal section as indicated by the line on its 
face. What angle does it make within the direction of vibration 
of the light incident upon it? Now turn the quarter-wave 
plate through 45 deg. Upon rotating the nicol, what happens 
to the transmitted light? Does your observation confirm the 
statement that a quarter-wave plate, with its principal section 
at 45 deg. to the plane of vibration of plane-polarized light 



398 LIGHT: PRINCIPLES AND EXPERIMENTS 

incident upon it, changes the plane-polarized to circularly 
polarized light? 

9. Replace the quarter-wave plate with a small sheet of 
cellophane. What do you conclude regarding the optical char- 
acteristics of cellophane? Wrinkle the cellophane and observe 
what happens to the transmitted light. What precautions does 
this suggest in the use of wave plates? Replace the monochro- 
matic source with white light, and repeat the observations with 
cellophane. 

10. Replace the cellophane with a cube of ordinary glass 
about 1 cm. on a side. Are there any variations in the light 
transmitted through different parts? Now put a small labora- 
tory clamp on the sides of the cube, and squeeze it. Observe 
that the effect of strains in an optically isotropic medium is to 
render it anisotropic. 

11. Replace the glass cube with a section of calcite cut per- 
pendicular to the optic axis. Observe the change in the pattern 
as the analyzer is rotated. Replace the white light with mono- 
chromatic light and repeat these observations with the section 
of calcite. Keeping in mind that the light is divergent, explain 
the rings and brushes seen. Because many nicols have too small 
a field, this experiment with the calcite section may be performed 
more easily with disks of polaroid used as analyzer and polarizer, 
instead of the polariscope. 

12. Use a sodium source. Remove the calcite crystal, sot the 
analyzer for extinction, and replace the calcite with a section of 
quartz about 5 mm. thick cut perpendicular to the optic axis. 
The light will reappear. Record the position of the analyzer 
and turn it to extinction once more. Does the angle through 
which it was turned confirm the statement that light transmitted 
along the optic axis of quartz has its plane of vibration rotated 
an angle to 21.7 deg. for every millimeter of thickness? Keep 
in mind that both right- and left-handed-rotatory quartz exist. 
Which is the specimen you have used? 

13. Make either a half-wave or quarter-wave plate of mica 
or cellophane and submit it to the instructor for approval. Since 
X5461 is so universally used, it is better to use the mercury 
source. If mica is used, it may be split with a fine needle. If 
cellophane, use the thickness ordinarily used for cigar wrappers. 
More than one layer may be necessary. Two with the directions 



EXP. 17] ANALYSIS OF ELLIPTIC ALLY POLARIZED LIGHT 309 

of the striae at an angle of about 45 deg. make a fairly satisfactory 
quarter-wave plate. 

EXPERIMENT 17 

ANALYSIS OF -EUBJByjLIiPOLARIZED LIGHT 
WITH AT QUARTER-WAVE PLATE 

The theory of elliptically polarized light and of wave plates 
contained in Sec. 13-11 should be read carefully before this 
experiment is begun. 

Apparatus. A spectrometer equipped with two graduated 
circles to fit over the collimator and telescope lenses, a third 
circle to be mounted as described later, a quarter-wave plate, 
preferably for Hg-5461, a Wollastori prism, two nicols, a Gauss 
eyepiece, and a mercury arc with a filter transmitting only the 
green line at 5461 angstroms. If a spectrometer table is not 
available, the essential elements of a collimator and telescope 
may be clamped on some suitable mounting or optical bench, 
since the experiment is performed with no deviation of the beam. 
The description following will assume the use of a spectrometer. 
Polaroids may be used instead of nicols, provided they are 
mounted in good quality optical glass. 

The experiment consists of producing a beam of plane-polarized 
light with a nicol and changing this to elliptically polarized 
light with a thin sheet of mica; then analyzing the light thus 
produced, to find the orientation and eccentricity of the ellipse. 
To this end, it is desirable to mount all the polarizing parts in 
the space between the collimator and telescope lenses. One of 
the circles can be clamped over the telescope lens, one over the 
collimator, and the third attached either to the telescope or 
collimator tube by an arm, as shown in Fig. 1, Exp. 18. * 

The Wollaston prism is used to accomplish the preliminary 
orientation of the nicols. With the collimator and telescope in 
line, and the slit vertical, place the Wollaston in the circle over 
the collimating lens, and rotate it until the two images of the 
slit are superposed in a vertical line. Now clamp a circle over 
the telescope lens, fasten its index at zero and put in it a Nicol 

1 If space prevents this, the middle circle may be over the telescope 
lens, and the third circle be placed at the eyepiece end of the telescope, hut 
this is not so desirable. If it is so done, the nicol in the third circle may be 
put between the field and eye lenses of the eyepiece. 



400 LIGHT: PRINCIPLES AND EXPERIMENTS 

prism with its plane of transmission approximately horizontal. 
This plane may be previously found by extinguishing, with the 
nicol, skylight reflected at the angle of complete polarization 
(about 57 deg.) from a plate of glass, the plane of transmission 
then being vertical. With the index clamped, turn the nicol 
slightly until one of the images of the slit is extinguished. Deter- 
mine this position as accurately as possible. Replace the 
Wollaston with this nicol, having clamped the index at zero, and 
turn it in the tube until the other image is extinguished. The 
planes of transmission of the polarizer and analyzer are now 
respectively vertical and horizontal. Keeping them crossed, 
mount the A/4 plate in the third (middle) circle, and turn it until 
the light is extinguished once more. Retaining all positions, 
fasten firmly over the end of the polarizer a thin piece 4 of mica. 
This may be done with a little wax or plasticine. The light will 
be restored, indicating that the mica transmits elliptically 
polarized light. There is, of course, the possibility that the 
mica may be a quarter- or half-wave plate, but this is not probable 
and may be guarded against beforehand. Now proceed as 
follows : 

a. Set the analyzer at the position for minimum intensity. 

6. Turn the X/4 plate through about 5 deg. and reset the 
analyzer for minimum intensity. This intensity will be either 
greater or less than that in (a). If it is less, the X/4 plate was 
turned in the right direction. If it is greater, then 

c. Repeat (6), turning the X/4 plate in the other direction and 
find a position of less minimum intensity than in (a). In any 
case, 

d. Orient the X/4 plate and analyzer to the positions for which 
extinction occurs. Call a the total angle through which the 
analyzer has been turned from its position before the mica wa? 
introduced, and call ft the corresponding total angle through 
which the X/4 plate has been turned. The axes of the ellipse 
are at the angle with that of the planes of transmission of the 
polarizer and analyzer when crossed. If the slit of the spectro- 
graph is accurately vertical and the preliminary adjustment with 
the Wollaston was accurately made, these planes of transmission 
are respectively vertical and horizontal. 

The ratio of the amplitudes in these directions is given by 
tan (a - 0). 



EXP. 



THE BABINET COMPENSATOR 



401 



Make a graph of the elliptical vibration, orienting the ellipse 
with reference to the planes of transmission of the nicols as X- 
and F-axes. 

Answer these questions : 

1. Suppose the mica had been a quarter- wave plate; could the experi- 
ment have turned out as described? Explain. 

2. Suppose it had been a half-wave plate; then could the analysis of 
elliptically polarized light have been carried out? Explain. 

EXPERIMENT 18 

THE BABINET COMPENSATOR 

Read carefully Section 13-12. 

Apparatus. A polarizer and analyzer, a Babinet compensator 
of the Jamin type, a quarter-wave plate, a spectrometer, a white- 
light source, and a source of monochromatic light such as a 
mercury arc with a filter to transmit 5461 angstroms. If a 




B 



$ 






1 


n 







;n 








II 


( 1 





I 

t 








II 
h 


j 
1 




.1 








M 
II, 


1 

L. 












111 

1 


i 










/ 


y 







FIG. 1. The Babinet-Jamin compensator B and analyzing nicol A, mounted at 

the eye end of the telescope tube. 

spectrometer is not available, a pair of ordinary laboratory 
telescopes may be used, if the experiment is one in which the 
state of polarization to be examined is produced by transmission 
as through a quarter-wave plate. In any experiment in which 
an angle of reflection is to be measured, the spectrometer is 
essential. For convenience, the instructions Will assume the use 
of a spectrometer. 

The polarizer is mounted at the slit end of a spectrometer 
collimator. At the eyepiece end of the telescope is mounted the 
Babinet compensator. Beyond this, as shown in Fig. 1, is 
mounted the analyzer. The eye end of the analyzer should be 



402 LIGHT: PRINCIPLES AND EXPERIMENTS 

equipped with a simple magnifyer of such focal length that in 
sharp focus are the cross hairs of the Babinet compensator. In 
some cases, instead of cross hairs, there are ruled two parallel 
lines on one of the wedges, perpendicular to the long edge of the 
wedge. The telescope and collimator should be adjusted for 
parallel light so that the image of the polarizing nicol will also 
be at the focal plane of the magnifyer. 

It is supposed that light in some unknown state of polarization 
is to be examined. For the purpose of class room experiment, 
this may be produced by using a X/4 plate oriented to such an 
angle that the light is elliptically polarized. The collimator 
and telescope should in this case be clamped in line. 

Part A. To Find the Phase Difference between the Com- 
ponents of the Elliptical Polarization. The instrument must first 
be adjusted. Use a source of white light (the unfiltered mercury 
arc will do in this case). Remove the compensator, and set the 
analyzer for extinction. The compensator is equipped with two 
adjustments: (a) One wedge may be moved with respect to 
the other; (6) the entire compensator may be rotated about the 
optical axis of the instrument. Rotate the compensator so that 
the pointer is on one of the 45-deg. marks, and replace it so that 
the wedges are parallel or perpendicular to the plane of trans- 
mission of the analyzer. This will be the case when the fringes 
disappear. Clamp the* compensator tightly on the draw tube 
and rotate the wedge 45 deg. This will give the position of 
maximum distinctness of the fringes. Place the cross hairs on 
the central fringe which is black. Replace the white light with 
monochromatic light. Now move the wedge until the next black 
fringe is under the cross hairs and record the distance moved. 
This distance may be called 2s corresponding to a difference of 
phase of 2ir and a path difference of X. Now introduce a plate 
of mica or a X/4 plate between the polarizer and analyzer so that 
the light incident on the compensator is elliptically polarized. 
In general the axes of the ellipse will not be parallel to the 
principal section of either wedge, and the fringes will be shifted. 
But it should be made certain that the polarization of the light 
transmitted by the X/4 plate is elliptical and not circular or 
plane. Move the wedge until the dark fringe is once more under 
the cross hairs. Calling the distance moved x, we have the 
corresponding difference of phase 



EXP. 1] ROTATORY POLARIZATION 403 






This is, however, the amount by which the phase difference is 
changed by passage through the compensator (see eq. 13-7). 
Hence 

2?rA = T-- (1) 

* v 

Part B. To Find the Position of the Axes of the Ellipse. Let 

plane-polarized light fall on the compensator and move the wedge 
through a distance s/2, having previously calibrated the microm- 
eter driving the wedge by measuring the distance 2s between the 
dark fringes. Then the cross hairs will be over a position at 
which the phase difference is ir/2, corresponding to a retardation 
of X/4. Now let the elliptically polarized light fall once more 
on the compensator. In general the middle black band will not 
be under the cross hairs, but it may be brought there by rotating 
the compensator. It will usually be necessary to rotate the 
analyzer also, to obtain maximum distinctness of the fringes. 
The axes of the elliptically polarized light are now parallel to the 
axes of the wedges. 

Part C. The Ratio of the Axes of the Ellipse. The situation 
will now be as shown in Fig. 13-21. OA and OB are parallel 
to the axes of the two wedges, OC is the direction of the principal 
section of the analyzer, and the direction of vibration of the 
light which is extinguished at the central fringe is DD' . If the 
analyzer is rotated through the angle 6, the fringes will disappear, 
since for this position the compensator will act like a quarter- 
wave plate. The tangent of 6 will be the ratio of the axes of the 
elliptical polarization. In the illustration the longer axis is in 
the direction OA. , 

r # 

EXPERIMENT 19 

ROTATORY POLARIZATION OF COMMON 

SUBSTANCES 

Read Sees. 13-19 and 20. 

There are many polarimeters designed solely for the measure- 
ment of the rotatory powers of optically active substances, 
especially for the measurement by this means of the purity of 
sugar. When the graduated circle of the analyzer is calibrated 



404 LIGHT: PRINCIPLES AND EXPERIMENTS 

in terms of purity instead of degrees the instrument is called a 
saccharimeter. In the absence of a special instrument, a polarim- 
eter may be constructed by arranging the necessary optical 
parts as shown in Fig. 1, which depicts essentially the Laurent 
polarimeter. 

M is a filter which gives approximately monochromatic light, 
P is the polarizing nicol (or polaroid), A is the analyzer, mounted 
in a graduated circle. At W is located some half-shade device 
such as the Laurent half-shade plate described in Sec. 13-23. 
With this device, the settings of the analyzer are made not for 
extinction but by turning the analyzer to the angle where the 
two halves of the field are of equal brightness. Accordingly, W 
M P W T A 



/V /! 


/ 


\ / f K 


/ \/ ! - 


l_ 


\y i_J * 



FIG. 1. The arrangement of the Laurent polarimeter. 

should be mounted so that a slight adjustment of its angle about 
the axis of the instrument should be permitted, in order to 
obtain sufficiently high brightness in the field of view. The 
ocular is really a very short focus telescope focused on the 
dividing line of the Laurent half-shade plate. At the position T 
may be placed either a tube or cell of solution, or a crystal 
specimen whose rotatory power is to bo determined. 

If the green line of mercury is to be used for a source, M may be 
omitted. If sodium light is used, M should be a cell with flat 
sides containing a solution of potassium bichromate, which 
stops the blue ordinarily found in such a source. 

A very bright sodium source is obtained by training the flame 
from an oxygas glass blowers' torch upon a small piece of pyrex 
tubing supported horizontally on a steel rod. For moderate 
intensity, the new General Electric sodium lamp may be used. 
The use of Hg-5461 or Na-5893 should depend upon the range 
of wave-length for which the Laurent half-shade plate is suitable, 
and for what wave-lengths the known values of specific rotation 
are available. Since the making of a half-wave plate from mica is 
within the skill of the average student, it is suggested that 5461 be 
used wherever possible. 

With the source and filter in place, focus upon the division 
in the half-shade plate and adjust the latter slightly until, with 
the analyzer set for equality of the two halves of the field, the 



EXP. 191 ROTATORY POLARIZATION 405 

greatest brightness is obtained. At least five settings of the 
analyzer for equality should be recorded, and the average taken. 
The analyzer should then be rotated through 180 deg. and the 
process repeated. These two settings 180 deg. apart are the zero 
settings of the polarimeter. Then the various specimens to be 
tested may be inserted at T and the analyzer rotated until 
equality of the two halves is once more attained. Each deter- 
mination of this position should be made at least five times, and 
the average recorded, repeating after rotation through 180 deg. 

It should be remembered that the insertion of a medium at T 
changes the optical path and throws out of its focus upon the 
dividing line of W, so that refocusing is necessary. 

Sometimes the angle of rotation is so large that it is not 
immediately known whether the rotation of the plane of vibration 
is clockwise or counterclockwise. A second observation may be 
made with a shorter length of substance or, in the case of solu- 
tion, with smaller concentration, to determine this point. Or, 
the rotation may be observed with both the 5461 line of mercury 
and the 4356 line. The rotation of the plane is usually smaller 
for the red than for the blue. 

Part A. The Optical Rotation of Quartz. The rotation is 
proportional to the thickness of quartz traversed (in the direc- 
tion of the optic axis), and depends upon the wave-length and 
slightly upon the temperature. If the angle of rotation p for the 
green mercury line is known, that for sodium light may be 
found from the equation 

<P(6893) = V>(5461) X 0.85085. (1) 

The temperature effect is given by 

Vt = *>o(l + 0.0001440 between 4 and 50C. 
The value of ^(54i) at 20C. for a plate 1 mm. thick cut perpen- 
dicular to the optic axis is 25.3571 deg. of arc. 

It should be kept in mind that quartz occurs both right-handed 
and left-handed. The specimen used should be examined with 
white light between crossed nicols to make sure that it is not cut 
from a twinned crystal. 

Part B. The Rotatory Power of Pure Cane Sugar. In this 
case the so-called direct method may be used, which supposes 
that the sample of sugar contains no impurities which are also 
optically active. 



406 LIGHT: PRINCIPLES AND EXPERIMENTS 

With a good analytical balance weigh out 26 gm. of pure cane 
sugar (rock candy), which has been previously pulverized and 
dried in a desiccator for about 24 hr., or over sulphuric acid 
in vacuo. Mix this thoroughly with distilled water, allowing 
none of the sugar to be lost, in a graduate to make exactly 
100 c.c. The most scrupulous cleanliness and exactness in 
measurement should be observed. Keep the solution covered 
to avoid loss by evaporation. 

Fill a tube 20 cm. long with the sugar solution, having first 
determined exactly what the length of the column of liquid will 
be. Record the mean of several determinations of equality in 
the two halves of the field. Repeat after rotating the analyzer 
through 180 deg. The specific rotation is given by 



where X is the wave-length used, a is the observed rotation in 
degrees of arc, I the length of the solution in decimeters, and c 
the concentration in grams per 100 c.c. of solution. The value 

[~\ 20C. 
a ' for cane sugar (sucrose, C^H^On) is 66.45 dee. 
J5893 

of arc. Since it is a little difficult to make observation at 
exactly 20C., a correction of 0.02 deg. of arc may be subtracted 
from the experimentally determined value for each degree centi- 
grade above 20C. 

The rotation ratio for sucrose, analogous to that given in 
eq. 1 for quartz, is 

? ( U9J) = *> ( M6i) X 0.84922. (3) 

Part C (Optional). The purity of a commercial sample of 
cane sugar may be tested by the method outlined in Part B. 
The sample should be prepared in the same manner as in Part B, 
except that the solution should be a little stronger than 26 gm. 
to each 100 c.c. of water, until it is certain that there will be 
no turbidity. If the solution is slightly turbid, it may be 
clarified as follows: Make a saturated solution of alum in 
water. Pour into about two thirds of it a slight excess of 
ammonium hydroxide and then pour in enough of the remaining 
one third to get a slightly acid reaction. Add to the sugar 
solution only a drop at a time. Too much will, because of the 



EXP. 20J VERIFICATION OF BREWSTER'S LAW 407 

change in concentration, introduce an error which cannot be 
neglected. The remaining water should now be put in to bring 
the concentration to specifications. 

Substituting the value of a obtained for this sample into eq. 2, 
calculate the value of c, and the percentage of purity. 

Part D (Optional). The methods described above make no 
provision for the errors possible due to the presence of optically 
active impurities. For this purpose, the invert method is used. 
Take 100 c.c. of the solution prepared in Part C, and add drop 
by drop 10 c.c. of concentrated hydrochloric acid, specific gravity 
1.2 (38.8 per cent solution), shaking meanwhile. Since the reac- 
tion is delayed, set this aside for not less than 24 hr. and keep the 
temperature at 20C. or over. Dilute the invert solution to 200 
c.c. Measure the rotation as before, and multiply it by 2, 
because of the reduced concentration. Calculate c as directed 
in Part C. 

The result obtained in this manner will be more accurate than 
by the direct method, provided the invert solution is properly 
made. 

Part E (Optional). Measure the rotations of several optically 
active substances. 

EXPERIMENT 20 
VERIFICATION OF BREWSTER'S LAW 

From Fresnel's laws of reflection, in Sees. 13-13 and 14, which 
should be read carefully before this experiment is begun, it 
follows that for transparent isotropic media tan i p = n, an 
equation known as Brewster's law. The angle i p is that for 
which i + T = 90 deg. This affords an experimental method of 
determining n, the principal difficulty being that it is required 
to find the angle of a Nicol prism, or other analyzer, at which 
extinction of a polarized beam occurs. 

It is also possible to measure the change, upon reflection, of 
the direction of vibration of a plane-polarized beam. This 
change is given by eq. 13-18: 

cos (i r) 

tan a jr. ( = tan /?, 

cos (i H- r) 

where a is the angle between the direction of vibration of the 
incident light and the plane of incidence and the angle between 



4Q8 



LIGHT. PRINCIPLES AND EXPERIMENTS 



the direction of vibration of the reflected light and the same 
plane. If both i and r are known, the values of ft for several 
values of a may be calculated. It is a simple matter to measure i 
and a, but the measurement of r is difficult. If n is known 
beforehand from some other experiment, r may be calculated 
from Snell's law. For this reason it is convenient to use in the 
experiment a prism or glass block for which the value of n has 
been quite accurately determined. 

Apparatus. A spectrometer equipped with graduated circles 

which may be clamped on the telescope and collimator tubes, a 

100 




30 40 50 60. 70 
Angle of Incidence, i 

Fio. 1. 



80 90 



pair of nicols to go in the circles, a Gauss eyepiece, sources of 
white and monochromatic light, and a prism or block of glass 
whose index is known. Polaroids may be used instead of nicols, 
but they should be mounted in good optical glass. 

Part A. This consists of a trial determination of n from 
Brewstcr's law. Adjust the collimator and telescope for parallel 
light, and clamp over the telescope lens one of the graduated 
circles containing a nicol. Open the slit wide. The circle and 
nicol should be previously examined so that the setting corre- 
sponding to the plane of vibration of the transmitted light is 
known, or a Wollaston prism may be employed as described 
in Experiment 17. Use a monochromatic source of as high 



EXP. 20] VERIFICATION OF BREWSTER'S LAW 409 

intensity as possible and one for which n is known. On the 
spectrometer table set the glass prism or block so that the light 
from the collimator is reflected from one face. It is important 
that the face be absolutely clean. The riicol should be set so that 
its plane of transmission is parallel to the plane of incidence, i.e., 
in a horizontal plane. This will insure maximum intensity at 
the beginning of the experiment. Rotate the spectrometer 
table slowly and the telescope twice as fast, so as to keep the 
light in view at all times. Somewhere in the neighborhood of an 
angle of incidence of 57 deg., depending on the glass used, it 
should be possible to extinguish the light by turning the nicol. 
Reference to Fig. 1 shows that this point of extinction is difficult 
to determine accurately, because the curve of reflection is not 
symmetrical about the point where 7 = 0. By means of the 
Gauss eyepiece find the normal to the glass surface. Substitute 
the value of i p in Brewster's law and compare the calculated and 
known values of n. 

Part B. Fit the second nicol into a graduated circle and clamp 
it over the collimator lens. Orient the nicol so that its plane of 
transmission makes an angle other than or 90 deg. with the 
plane of incidence. Use monochromatic light for which n is 
known. With the analyzer measure for several values of i 
on either side of the angle of complete polarization determined 
in Part A. The angle i should be at intervals of between 2 and 
5 deg. To find each 0, the analyzer should be set for extinction 
several times and a mean value recorded. Do not forget to 
record the position of the normal at each setting i. (To render 
this unnecessary, it is possible to use a very convenient table, so 
geared that it turns half as fast as the telescope. Ordinarily 
this is called a minimum deviation table. With this equipment, 
i need only be measured once.) Make a table of data, with i in 
the first column, calculated in the second, and measured /3 in 
the third. There should, of course, be a change in the sign of 
at the angle of complete polarization. Plot the measured 
angles of against the corresponding values of i. The point 
where the curve crosses the i-axis gives i p . From this, n may 
be calculated. 

How does the value of n obtained compare with that from 
Part A? with that from other experiments? What are the chief 
sources of error in this experiment? 



410 LIGHT: PRINCIPLES AND EXPERIMENTS 

EXPERIMENT 21 
THE OPTICAL CONSTANTS OF METALS 

The theory of metallic reflection is discussed briefly in Sees. 
15-10 and 11. For a more extended discussion the student 
should consult Drude, "Theory of Optics," and Wood, "Physical 
Optics." The principal experimental facts may be summarized 
as follows: 

a. Metals do not completely polarize light at any angle of reflection. 

6. Plane-polarized light incident upon a metallic surface is upon 
reflection changed to elliptically polarized light, unless the plane of 
vibration is either parallel or perpendicular to the plane of incidence. 

c. The ellipticity is due to a phase difference A introduced, on reflec- 
tion, between the components of the vibration parallel and perpendicular 
to the plane of incidence. The ellipticity is greatest for the angle of 
incidence <p for which this phase difference A = ir/2. This angle is 
called the angle of principal incidence $. 

d. Circularly polarized light incident at the angle <p is reflected as 
plane-polarized light with its plane of vibration making an angle ^ with 
the plane of incidence. The angle $ is called the angle of principal 
azimuth. 

As stated in the text, theoretical relationships can be obtained 
between these angles and the quantities n, the index of refraction 
of the substance, and K, the absorption index. These are given 
in one form with a high degree of approximation in eqs. 15-15, 
for any value of A. In order to make these equations applicable 
to the experimental conditions described above, A is put equal 
to IT/2, whereupon eqs. 15-15 become 

K = tan 2#, 
w 2 (l + * 2 ) = sin 2 cos 2 . 

As will be seen from (c) and (d} above, there are two experi- 
mental methods of determining n and K. Incident plane-polar- 
ized light may be reflected from the surface at the angle <p, and 
the value of # determined for which it becomes circularly polar- 
ized upon reflection. Or, incident circularly polarized light may 
be reflected as plane-polarized and the azimuth # determined 
for which it is completely extinguished by an analyzer. 

Since difference of phase is responsible for the change of polar- 
ization, the experiment should be performed, not with white, but 
monochromatic light. 



EXP. 21] THE OPTICAL CONSTANTS OF AfETALS 411 

With n and * known, the value of the reflectivity R may be 
calculated by means of eq. 15-10. 

Apparatus. The spectrometer, polarizer, and analyzer used 
in Experiments 17 and 20, or, the polarizer, analyzer, and Babinet 
compensator used in Experiment 18. A mercury arc and filter 
for X5461. Several plane glass surfaces freshly coated with 
metals. It is recommended that heavy opaque aluminum, 
silver, and copper be deposited by evaporation on plane glass 
surfaces about 2.5 cm. square. If evaporation is not possible, 
polished plane surfaces may be used, although in the polishing 
process the surface often takes on foreign matter which changes 
its character. 

Part A. n and K with the Babinet Compensator. Set the 
wedges of the compensator so that the central dark fringe is dis- 
placed through a distance which corresponds to a phase differ- 
ence of one quarter of a complete period. Set the analyzer for 
maximum blackness of the fringe. Then allow plane-polarized 
light to fall on one of the coated plates, which should be held 
securely at the middle of the spectrometer table. Change the 
angle of incidence by rotating the table, and the telescope twice 
as fast, until the central dark fringe comes back to the central 
position. Record this angle of incidence, which is <p. Then 
turn the analyzer until the central fringe is once more black. 
This will give the angle #. 

Part B. Alternative Method with Quarter -wave Plate. With 
a X/4 plate change the plane-polarized light transmitted by the 
polarizer to circularly polarized light. Let this be incident upon 
the metallic surface, and find the angle of incidence (<p) for which 
the reflected light is plane-polarized, as determined by the 
analyzer. Record also the azimuth of the analyzer, i.e., the 
angle its plane of transmission makes with the plane of incidence. 

Substitute the values of <p and # found by either of these 
methods in eqs. 1 to get n and K. 

From eq. 15-10, calculate R, the reflectivity. The following 
values of n and K are taken from the International Critical Tables, 
Vol. V, page 248. They are for opaque surfaces. Semitrans- 
parent surfaces yield different values. The values of the optical 
constants of metals vary widely among observers, principally 
because of the difficulty of obtaining an uncontaminated 
surface. 



412 



LIGHT: PRINCIPLES AND EXPERIMENTS 





X 


n 


K R calc. 

i 


Silver 


4500 


0.16 


! 
14.5 88. 




5000 


0.17 


17.1 90. 




5500 


0.18 


18.8 


91.5 




5893 
5893 


0.18 
0.20 


20.6 { 
17.1 ( 


Different observers 




6000 






92.7 




6300 


0.20 


19.5 


1 


Aluminum 


4310 


0.78 


2.85 






4860 


0.93 


3.15 






5270 


1.10 


3.39 


* 




5893 
5893 


1.44 
1.28 


3.64 J 
3.66 { 


Different observers 




6570 


1.48 


3.92 




Copper 


5000 


1.17 


2.03 






5600 


0.855 


2.83 






5893 
5893 


0.62 
0.64 


4., t 
4.08 I 


Different observers 


0000 


0.565 


5.51 





EXPERIMENT 22 
POLARIZATION OF SCATTERED LIGHT 

Read Sees. 15-12 and 13 on the scattering of light and its 
polarization. 

Apparatus. A spherical liter flask, a 500-watt projection 
amp, a cylindrical shield of metal with a slot in one side about 
I cm. wide. A Nicol prism or polaroid in a graduated circle. 
k little hyposulphite of soda and some concentrated sulphuric 
icid. 

Clean the flask and put in it about a quarter of a teaspoonful 
)f hyposulphite of soda. Half fill the flask with distilled water 
ind shake it to dissolve the hyposulphite of soda and also to 
5et rid of the bubbles which gather on the sides. In a graduate 
nix about 0.5 c.c. of concentrated sulphuric acid with 100 c.c. 
>f distilled water, and wait till it clears. Set the lamp in a 
vertical position, base down, put the shield around it, and set 
.he flask so that the broad wedge of light through the slot falls 
m the hyposulphite of soda solution. 



EXP. 22) POLARIZATION OF SCATTERED LIGHT 413 

Mount the nicol (or polaroid) so that it may be swung about 
a vertical axis permitting analysis of the state of polarization 
of the solution at any angle. For this purpose the flask may be 
set on the table of a spectrometer, and the nicol and graduated 
circle on a short tube put in place of the telescope. This is not 
necessary, since the nicol may also be mounted on a stand which 
can be shifted from one position to another. 

With the analyzer, observe the scattered light perpendicular 
to the beam and estimate the amount of polarization. Make the 




FIG. 1. 



same estimate of the polarization in the direction (nearly) of the 
beam. The light will be too bright to look at directly, and it 
will be best to look instead at an angle of about 20 deg. with the 
direct beam. Repeat these observations with a red filter in the 
path of the beam. 

Pour in the dilute acid and give to the flask a slight rotating 
motion to produce a vortex in the liquid. If this is done 
successfully, the precipitation will take place mostly in the center 
of the flask, as shown in Fig. 1. 

With the nicol set so as to observe the liquid at right angles to 
the beam, note the increase of polarization as precipitation 
increases. Occasionally make the same observations at other 
angles to the beam. 



414 LIGHT: PRINCIPLED AND EXPERIMENTS 

Write a description of what you have observed, noting espe- 
cially the degree of polarization before and after the acid was 
poured in the flask, the growth in polarization, the direction of 
vibration of the scattered light, the color effects, and any other 
effects you have seen. 

EXPERIMENT 23 

THE FARADAY EFFECT 

For the theory of this experiment, read Sec. 16-6. 

Apparatus. In order to produce the rotation of the plane of 
vibration of a light beam traversing a medium in a magnetic 
field, it is necessary to have a coil of considerable field strength. 
While such coils are not ordinarily part of the equipment of a 
light laboratory, they actually cost far less than many pieces of 
optical equipment and may be used for a variety of purposes, in 
laboratory instruction, lecture demonstration, and research. A 
coil with a hollow cylindrical center, having a field strength of 
about 1,000 gauss, will produce a measurable Faraday -effect 
in a column of carbon bisulphide 15 cm long. The field at the 
center of a single solenoid of length L and radius jR, due to current 
7 through n turns per centimeter, is given by 

H = 4irnl . J -;.=) (gauss) 

+ R 2 



and for many layers of turns, there will be a value of.// for each 
layer. For successful operation over any but very short periods 
of time, the coil should be water-cooled. There are different 
systems of water cooling, a common one being the insertion every 
5 or 10 cm. along the coil of a hollow disk through which water 
flows at a good rate. Unless the laboratory is equipped for 
the construction of a properly insulated and water-cooled coil, it is 
wiser to purchase one of sufficient strength, exactly as telescopes, 
spectrometers, microscopes, and other accessories are purchased. 
The remaining items of apparatus consist of a polarizer and 
analyzer, a graduated circle for the latter, a half-shade plate, a 
mercury arc and filter, or a sodium source. The use of sodium 
is not recommended as more precise determinations are possible 
with Hg-5461. Tubes for containing liquids may be similar 
to those used for the sugar experiment, Experiment 19, Part B. 



BXP. 23] THE FARADAY EFFECT 415 

A satisfactory tube can be made by cutting a heavy-walled tube 
of pyrex to the appropriate length with a hot wire and grinding 
the ends flat. Or, if glass-blowing equipment is available, the 
ends of a section can be moulded into a flange about 5 mm. 
wide which can be ground flat. After grinding, on the ends 
should be sealed circular disks of good quality glass. A good 
sealing material which is acid-proof arid impervious to ordinary 
solvents is Insa-lute Adhesive Cement. If it is desired to use a 
separate tube of some permanence for each liquid, one end disk 
may be sealed on, given time for the cement to set, then, with 
the tube in a vertical position, liquid may be poured in almost 
to the top of the tube, and the other end sealed on. The small 
fintount of air thus enclosed will not cause any difficulty. If the 
tube is to be emptied and refilled, it should have a side opening 

* 

M 

ooopoooooooooooooooooooooooooooooooo 






oooooooo.ooooooooooooooooooooooooocx>ol 



Fio. 1. 

which can be corked up, but which snould not be so long as to 
interfere with the insertion of -the tube in the coil. Great care 
should be taken to avoid the ignition or explosion of any volatile 
liquid used. 

Part A. Arrange the apparatus as shown in Fig. 1. Since 
the field is not uniform at the ends of the coil, it is suggested 
that the coil be about 30 cm. long, with a hollow cylindrical center 
about 5 or 6 cm. in diameter. The tubes of liquid should be 
about half as long. Since, however, thf experiment is not 
expected to produce rigorously accurate results, no harm is done 
if they are longer. 

Determine the rotation for carbon bisulphide. The Verdet 
constant R for carbon bisulphide is 0.0441. Substitute 6 and R 
in the equation - RIH (in minutes of arc) where / is the 
length of the liquid column in centimeters, and calculate H. 
Repeat for distilled water, whose Verdet constant is 0.0269. 

Part B (Optional). Measure the indices of refraction and dis- 
persion of carbon bisulphide in a hollow prism, with walls of good 



416 LIGHT: PRINCIPLES AND EXPERIMENTS 

optical glass free from strains, over the range from about 4300 to 
6000 angstroms by the method of Experiment 5. Find the 
value of dn/d\ for X4356 and X5461 of mercury, and for the 
sodium lines. Find the angle of rotation 6, and substitute 
0, X, dn/d\, e, m, and c in the equation 

e 



and thus calculate H. 

In each case a good filter is desirable, as the color effects due to 
dispersion of rotation are pronounced. 

Part C (Optional). If facilities permit, an exceedingly inter- 
esting experiment is to measure the rotation due to a semi- 
transparent iron film in a magnetic field. With the evaporating 
outfit (Appendix V) deposit a thin film of iron on one half of a 
circular disk of good optical glass about 1 in. in diameter, and 
free from strains. A coat which transmits between one half 
to one fourth of the light is satisfactory. Fasten this in a 
support which will hold it in a position normal to the field and 
place it at the middle of the coil. Measure the rotation produced 
by the glass alone and that by the iron coating and the glass, 
subtract the first from the second, to obtain the rotation due to 
the iron film. Since the angle will be very small, the experiment 
should be regarded only as qualitative, and a^ a demonstration 
of the rotatory power of iron. A field of 5000 gauss or over 
will be required for this experiment. 



APPENDICES 



APPENDIX I 



A COLLINEAR RELATION USEFUL IN GEOMETRICAL 

OPTICS 

For an ideal optical system, a point-to-point, line-to-line, and 
plane-to-plane correspondence between object and image may 
be expressed in terms of coordinate geometry by the equations 



= 



y = 



I 



ax 4- by 

4- 



-\- cz 
4- 



_ 

d 



ax 4- by + cz -\-d 
4" b s y -\- c*z 4- d s 

-f by 4- cz 4- d \' 



(D 



in which x, y, and z represent the coordinates of an object point 
and x', y f , and z f the coordinates of the conjugate image point. 
We may conventionally prescribe that the object space be 
placed on the left, with positive directions to the right and up, 
while the image space be on the right with positive directions to 
the left and up. 

This general relationship can be limited for ideal symmetrical 
coaxial optical systems. A symmetrical optical system is one 
in which each reflecting or refracting surface is generated by 
rotating an element of the surface about the optic axis. In such a 
system, for any plane of incidence containing the optic axis, the 
magnification in, and position of, the image plane will always 
be the same for a given object plane. Hence we need consider 
only the xy- and x'y '-planes, the optic axes being in the x- and 
^'-directions. Equations 1 may accordingly be simplified to 



x' = 



a\x 
ax 



4- 



4- by 

+ 



4- d 
4- dz 



ax 4- by + d 



(2) 



A second property of symmetrical optical systems is that a 
change in the size alone of the object causes no change in the 

419 



420 LIGHT: PRINCIPLES AND EXPERIMENTS 

position of the image, but only a conjugate change of size. This 
means that a change in the value of y in eqs. 2 must cause no 
change in x', but only in y'. As the equations stand, this is not 
true, but it can be made true by putting 6, 61, 02, and rf 2 each equal 
to zero, reducing the equations to 






ami *--,. (3) 



-, 

ax 4- d ax -f d 

These may be solved for x and y, obtaining 

di dx' , y f (adi aid) 

x = - ,, an( i y = v l ' 

ax a\ 



Now the coefficients a, d, a\, di, and b 2 are values for a particular 
optical system depending on the radii of curvature of the surfaces, 
their distance apart, and the indices of refraction of the media. 
Furthermore, the linear equation ax -f- d = obviously repre- 
sents the principal focal plane in the object space, since it places 
x' at infinity. Likewise the equation ax' ai = represents 
the principal focal plane in the image space. By the substitu- 

tions x = xi -- and x' = x\ H -> the origins are shifted to 

a a 

the principal focal points. The resulting equations are 

ad i aid 

(4) 



XiXi = 



y 



y ax i 
We may further simplify eqs. 4 by putting 



i - aid _ ff , , 62 

a* " ' a 



where / and /' are constants depending upon the radii of curva- 
ture of the surfaces, their distances apart, and the indices of 
refraction of the media, so that finally, dropping the subscripts, 

xx' = //', and y - = f - = y- (5) 

y x j 

Equations 5 hold for any ideal optical system in which the fore- 
going limitations placed on the general collinear relation exist. 



APP. TIJ CORRECTION FOR SPHERICAL ABERRATION 421 

APPENDIX II 

THIRD -ORDER CORRECTION FOR SPHERICAL 
ABERRATION FOR A THIN LENS IN AIR 

From Fig. 2-1, 

n' _ sin <p _ a -f- r\ b' 

n sin <?' b a' TI ^ ' 

But by the cosine law, 

6 2 = (a + r,)* + r, 2 - 2(a + rOn cos a (2) 

and 

6' = (a' ri) 2 + ^i 2 -f 2(a' ri)ri cos a. (3) 

a 2 a 4 a 6 
By expansion, cos a = 1 o7 + T]~ A! "^~ ' ' ' Neglecting 

higher powers of a than the second, and substituting for a its 
approximate value 



and 

6 = a (5) 

Using the binomial theorem 

{x ~T~ y) ~ x ~t~ tix y ~i~ 
for n 2, considering the right-hand member of eq. 4 to be 

CL I t*i h^ 

2xy, y = - -^ and, to a sufficient degree of approximation, 
ri 2 

considering h to be small. 



and, similarly, 

-<['--?)} 

Substituting eqs. 6 and 7 in eq. 1, 



422 LIGHT: PRINCIPLES AND EXPERIMENTS 

in which HZ and n\ have been substituted for n' and n, and the 
subscript m is used to indicate the image distance with a single 
surface. This substitution is made in order to facilitate the 
application of the results to the case of a lens in air, in which 
case n\ n$ = 1 and n z = n. 

Since in view of the approximation already made h is small 
with respect to the other dimensions, we can substitute for a' 
wherever it occurs in the coefficient of /i 2 the value derived from 
the first-order equation, 

n\ nz _ HZ n\ 
a a' r\ 

whence 

Hl n ' 2 n * ~ ni 



a a m ' 



4. _i_ 4- , . 



For the second surface of the lens, the distance to the virtual 
object from the vertex is a m ', hence the equation analogous 
to eq. 8 for this surface is 

_ n - j_ _^ n 3 ~ n i 
' ' ~ 



/i 2 / 



where a^ is the distance from the lens to the focus for the rays 
intersecting the lens at a distance h from the axis. Since h is 
small, we can substitute for a m f wherever it occurs in the coeffi- 
cient of A 2 the value of a m ' derived from 



a m ' a' r z 



whence for the second surface, 



__ ^ V 

a'n 2 /\ 



Apr. II] CORRECTION FOR SPHERICAL ABERRATION 423 

For the case of a lens in air, n\ = n 3 = 1 and n 2 = n. Substitut- 
ing these values in eqs. 9 and 11, 

1 n _n-\ h*(n - 1 V 1 1\V + 1 1\ 
a + 57 ~ ~TT~ + 2\~^~An + / \~^~ + n/ 



l \( 1 _ iVA 4- 1 _ J_\ 
/\^ a'/ \ ' r 2 / 



and 

_ JL + J_ = l - n . h *( n 
a m ' a k ' r 2 + 2\ n 2 

(13) 
The sum of eqs. 12 and 13 is 



n - 1 W\(l iV/n + 1 , A 
n 2 ' 2L\r, ^ a) \ a "*" n/ 




For paraxial rays, 

1 . l f ^( l l \ /i-x 

- H , = (n 1)1 --- ) (lo) 

a a' '\ri r 2 / v ' 

Hence, 

I _ J_ = _!L~J. ^ 
a' a k ' ~ n 2 " 2 ' 

where K is the quantity in the large brackets in eq. 14. The 

difference between the focal lengths for paraxial rays and those 
intersecting a lens a distance h from the axis may be written 

/ / ^ n I h~K , , 

a k ' - a' = -a' -^ --- =- (lb) 

11 & 

provided the difference between a k ' and a' is small enough so 
that for their product may be substituted a' . 



424 



LIGHT: PRINCIPLES AND EXPERIMENTS 



APPENDIX III 

DERIVATION OF EQUATIONS FOR ASTIGMATIC 
FOCAL DISTANCES AT A SINGLE REFRACTING 

SURFACE 

In Fig. 1 let be an object point, not on the axis, in the plane 
containing the line element AP of a single spherical refracting 
surface and C, its center of curvature. Then if coma is absent, 
all the rays which have the same inclination u as OP with OC 
will intersect the line OC extended in a point such as 1%. Let 




Fio. 1. 

OP = s, and PIz = s 2 . Then, from the law of sines, in triangle 
OP/2 

8 82 0/2 



sin u' sin u sin (<p <?') ' 



in triangle OPC 



r 



= = OC . 

sin a sin u sin v 5 ' 

and in triangle PC/ 2 

r ,S'2 C/2 



(1) 



(2) 



sin u sn a sn 



From eq. 2, 



from eq. 3, 



sin u 

_ r sin <p' 
2 sin u' 



(3) 



(4) 



(5) 



APP. Ill] EQUATIONS FOR ASTIGMATIC DISTANCES 425 

Adding eqs. 4 and 5, 



PC + C7. - 07. - - + (6) 

sin w sm w v ' 

Substituting this value of 0/ 2 in eq. 1 and using the first and last 
members of eq. 1, 



s 



sn w sn 



[shijp sin <p' 
sin u sin t*' 



(7) 



From the first and second members of eq. 1, sin u (s 2 sin u')/s, 
whence eq. 7 becomes 



s 



sn u sn 



I '" -" '1 

(<p </)LS2 sin u' sin u' \ 



Expanding sin (<p <?') and substituting for sin <p its value from 
Snell's law, i.e., 



sin <p = sin ^', (9) 



eq. 8 may be written 



n n _ n cos <f> n cos <p , . 

s s z r 

This gives the distance s 2 , measured from the surface, of the 
sagittal or secondary focus. 

Consider next two rays, OP and another adjacent ray OA. 
Since they are refracted by the surface at different distances 
from the intersection of OC with the surface they will, after 
refraction, intersect at a point I\ not on OC extended. Let the 
angle between OA and OC be u + du, that between AI\ and 
P/i be du', and let PI\ s\. Since fron>the figure 

<p a + w, and <?' = a u', 
by differentiation it follows that 

d<p = da + rfw, and d<p f = c/a' dw'. (11) 



Considering the angles du, du', and da to be equal to their sines, 
it follows from the law of sines that 

, PA cos <p t , PA cos <?' , PA M0 v 

du = ---- -t du - ----- > da = - j (12) 

' 



426 LIGHT: PRINCIPLES AND EXPERIMENTS 

whence, in eq. 11, 



/I 
I - 

\r 



T) ^ i cos j j / r>^ 

= P.4I - H -- - , and d< = PA 

s 




Differentiation of SnelPs law in eq. 9 gives 

n eos <p d^ = n' cos <p f d<p', (14) 

and on substituting the values of dy and dp' from eq. 13 this 
becomes 

1 COS < 



cos 



r s 



, ,[ 1 COS <p'l 

= n cos <p > 

L r si J 



which may be written 



n cos 2 <p n' cos- \p _ n' cos </?' n cos 



This gives the distance lf measured from the surface, of the 
tangential or primary focus. 

APPENDIX IV 

ADJUSTMENT OF A SPECTROMETER 

Spectrometers vary widely in their adaptability to a variety 
of uses, precision and ease of adjustment, and consequently in 
cost. For most of the experiments in this book a moderately 
expensive instrument will serve as well as the most costly to 
demonstrate the principles of optics. The precision to be 
desired is perhaps greatest in the case of the experiments on the 
index of refraction of a prism and the dispersion of prisms and 
gratings. There are certain minimum requirements to be met 
by any instrument. The optical parts should be of good quality, 
the mechanical construction should be rigid and sufficiently 
massive to prevent flexure, and the graduated circle should permit 
an accuracy of setting at least to 5 sec. of arc. 

The essential parts of a spectrometer are: A circle graduated 
in degrees of arc, equipped either with verniers or micrometer 
microscopes with which angles may be read; a rotatable table 
on which prisms or other optical parts may be set; a collimator 
and slit; and a telescope. The ideal arrangement is to have each 
one of these four parts independently rotatable on a cone or axis, 
but arranged so that the table, the collimator, and the telescope 



APP. IV] ADJUSTMENT OF A SPECTROMETER 427 

may either be clamped to the mounting or to the graduated 
circle. Sometimes the verniers (or microscopes) are fixed to 
the arm of the telescope so that they rotate with it, while the 
table, the collimator, or both, may be clamped to the graduated 
circle. In any case, the user of the spectrometer should study 
the demands made by any particular experiment before proceed- 
ing with its performance. 

When in adjustment, the telescope and collimator tubes should 
be set so that no matter what their angle about a vertical axis 
may be, their axes are always perpendicular to, and intersect, the 
main vertical axis of the spectrometer. A satisfactory instru- 
ment will be equipped with devices to make this possible. More- 
over, when in adjustment, the slit of the collimator should be 
at the principal focus of the collimator lens, and the telescope 
cross hairs should be at the principal focus of the telescope 
objective. 

1. Adjustment of the Telescope for Parallel Light. Method 
1. Remove the telescope from the spectrometer and point it 
at a bright object, such as a lamp globe or the sky. With the 
eye previously accommodated to distant vision, slide the eyepiece 
in or out in the draw tube, keeping the position of the cross 
hairs fixed until the cross hairs appear sharp. This will insure 
that the experimenter makes observations with a minimum of 
eyestrain. It is well, perhaps, as a final adjustment, to pull 
the eyepiece out to the point beyond which the cross hairs 
appear to become slightly blurred. Next point the telescope 
through an open window at some object a few hundred feet 
away, and rack the cross hair and eyepiece together in or out 
until a distinct image is seen. Try this several times. Make 
sure that the eyepiece has a sufficiently snug fit in the drawtube 
so that it will not slip too freely and destroy this adjustment. 

Method 2. This involves the use of a Gauss eyepiece, without 
which several experiments cannot be performed. It is described 
in detail in Sec. 7-3 and is illustrated in Fig. 7-4. Its relation 
to the spectrometer is illustrated in Fig. 1. Light from a source 
will thus be reflected past the cross hairs through objective O. 
If it is then reflected from a plane surface M, such as the face of a 
mirror or prism, directly back into the collimator, an image of 
the cross hairs will appear. Provided the cross hairs are at the 
principal focus of 0, this image will be in the same plane. When 



428 



LIGHT: PRINCIPLES AND EXPERIMENTS 



the cross hairs and their image are both in sharp focus, with no 
parallactic displacement, the telescope is in correct adjustment 
for parallel light. 

2. Adjustment of the Collimator for Parallel Light. The 
collimator and the telescope should next be set so that their 
axes are coincident and intersect the vertical axis of the spectrom- 
eter. This may first be done roughly by sighting along the 
tubes. A finer adjustment may be made by the use of a block 
set on the prism table with a vertical edge at the center of the 
table, sighting past it with -the slit and eyepiece removed, with- 
out disturbing the position of the cross hairs. Replace the 



Cross- 
hairs 



II 




Fia. 1. Sketch of a telescope equipped with a Gauss eyepiece 

eyepiece and slit, taking care to bring the former once more into 
correct focus on the cross hairs. Open the slit to a convenient 
width, say a millimeter or less. Rack the slit in or out until a 
sharp image of it is at the plane of the cross hairs without paral- 
lactic displacement. The collimator will then be adjusted for 
parallel light. 

2a. Alternative Method of Focussing a Spectrometer. Schus- 
ter's Method. If neither a distinct object nor a Gauss eyepiece 
is available, the following method, due to Schuster, may be 
employed. 

Use the mercury arc with filter for 5461 angstroms, or a sodium 
source. Adjust the telescope and collimator in a straight line 
across the center point of the spectrometer table. Put the 
prism so that it has maximum illumination from the collimator 
and orient it to the position of minimum deviation (see Sec. 8-1). 
Rotate the prism slightly to the other side of minimum deviation 



APP. IV] 



ADJUSTMENT OF A SPECTROMETER 



429 



following the image with the telescope, and focus the collimator 
for sharpest image. Repeat the alternations of rotating and 
focusing, first telescope and then collimator, until turning the 
prism causes nd change of focus. When this condition is reached, 
the rays from any point on the slit are parallel in passing through 
the prism. 

3. Adjustment of the Telescope so that Its Axis Is Perpen- 
dicular to the Axis of the Spectrometer. For this purpose it is 
desirable to provide a plane-parallel plate coated on both sides 
with a reflecting metallic surface. If a plane-parallel plate is 
not available, a plate with one side plane and metallically 
coated may be used instead. The plate should be mounted 
in a metal holder like that shown in Fig. 2, 
so that it may be set firmly on the spectro- 
meter table and the possibility of breaking 
may be minimized. If the base is made 
somewhat larger than shown, and three 
adjusting or leveling screws are inserted, 
its usefulness will be increased. Set this 
plate so that the telescope may be pointed 
to either face without interfering with the 
collimator or verniers. Illuminate the cross 
hairs by means of the Gauss eyepiece, and manipulate the telescope 
and "table until an image is reflected back into the field of view. 
At first this will be difficult as some experience is needed to insure 
good illumination of the cross hairs. A good procedure is to look 
directly into the mirror with the telescope swung to one side so 
that the image of the eye appears at about the same level as the 
center of the objective and at a point in the mirror directly 
over the center of the table. Then swing the telescope to 
position between the eye and the mirror. Then move the 
telescope from side to side slightly with different adjustment of 
the telescope leveling screws until a glimpse is caught of the cir- 
cular area of illumination reflected back through the telescope. 
When the image is located, it will probably be either too high 
or too low. Bring it into coincidence with the cross hairs by 
adjusting the telescope leveling screws for one-half of the correc- 
tion and the table leveling screws for the other half. Then 
rotate the telescope through 180 deg. until it is pointing to the 
other side of the mirror, and repeat the adjustment. After 




FIG. 2. 



430 LIGHT: PRINCIPLES AND EXPERIMENTS 

several corrections of position on either side, the image of the 
cross hairs should coincide with the cross hairs themselves, no 
matter on which side the telescope may be. It should be 
noted that, although the mirror surface is in adjustment, the 
table may not be, so that the substitution of a prism or grating 
may necessitate some further leveling. The telescope, however, 
should now be correctly set so that its axis is perpendicular to, 
and intersects, the principal axis of the spectrometer. 

In some cases the axis of rotation of the table is not coincident 
with the axis of the instrument. The adjustment above is, 
however, the most useful one. In case it is desired simply to 
adjust the telescope perpendicular to the axis of the table, this 
may be done by moving the mirror and table through 90 deg. 
between adjustments with the Gauss eyepiece. 

The collimator may now receive its final adjustment. Set the 
telescope and collimator in a straight line pointing toward a 
light source so that the slit, in a vertical position, coincides with 
the intersection of the cross hairs. Place a hair, toothpick, or 
fine wire across the slit and on a level with the center of the 
collimator tube, and adjust the leveling screws of the latter 
until the shadow of the obstacle coincides with the intersection 
of the cross hairs. Sometimes the slit length and eyepiece 
magnification are such that no obstacle is required, both ends 
of the slit being in view at the same time. 

An alternative method is to rotate the slit to a horizontal 
position for leveling the collimator. This is not generally 
recommended, since often there is no provision for free rotation 
of the drawtube of the slit. Forced rotation of the slit may then 
tend to destroy some defining pin in the tube, or wear the threads 
of the connection between the drawtube and the slit so that the 
latter may no longer be definable in a vertical position. 

APPENDIX V 
PREPARATION OF MIRROR SURFACES 

1. Chemical Deposition of Silver. For this method the stu- 
dent is referred to the "Handbook of Chemistry and Physics," 
published by the Chemical Rubber Publishing Co. The method 
is tedious and uncertain and should not be used unless the more 
satisfactory evaporation method cannot be used. 



APP. VJ 



PREPARATION OF MIRROR SURFACES 



431 



2. Deposition by Evaporation. This is by far the most 
satisfactory method, and can be used for the greatest variety 
of substances. Since the essential parts of the apparatus are a 
tungsten coil which can be raised electrically to a high tempera- 
ture, an enclosure in which the pressure can be reduced to 
approximately a cathode-ray vacuum, 
and a rack for supporting the plate to 
be coated, quite simple apparatus can 
be utilized. The writer has obtained 
good surfaces, for use in a small inter- 
ferometer, by making use of a liter 
flask into which was sealed temporarily 
a glass plug carrying the leads to the 
heating coil, the vacuum being obtained 
with an oil pump and a trap of outgassed 

charcoal. However, such devices are 

FT? 
only temporary, and the laboratory in L - 

which much optical work is to be done 
should be provided with a more perma- 
nent equipment, such as is illustrated in 
Fig. 1. 

A base plate of steel is mounted 
firmly on a stand, table, or rigid shelf. 
The base plate should be thick enough 
to withstand the force of atmospheric 
pressure on its lower surface, with a FlG i Evaporating a ppa - 

wide margin of safety. For a bell jar ratus. B, base plate; WW, 
_. ,. , ,, ! , , i i i_ water-cooled electrodes; T, 

of 6 in. diameter the plate should be u ^ d air or CO2 trap . D 
cold-rolled steel % in. thick, and thicker diffusion pump; F, to fore 

. , . mi i_ J.L i i pump; P, plate to be coated; 

for larger jars. Ihrough the plate are ^ shield . R> glass rod for 
drilled a hole about 1 in. in diameter moving shield; s, sylphon; E, 

_ .. j , ,1 testing electrode; C, heating 

for evacuation and two or more holes coiL 
about 3 in. apart for the terminals. 

While two terminals are sufficient for most purposes, three permit 
the use of two heating coils which may be used for different 
metals. The terminals should be water-cooled so as to prevent 
overheating, a suggested design being shown in Fig. 2. Ordinary 
automobile spark plugs screwed into the base plate from above 
have been used in place of water-cooled terminals, but they are 
short-lived. 




jrr . . ; A 

r -* ^ I 



432 



LIGHT: PRINCIPLES AND EXPERIMENTS 



The bell jar should be of good quality and ground with emery 
on the base plate. The neck should preferably be of the type 
which can be fitted with a stopper. On this is fastened with some 
suitable cement such as deKhotinsky, sealing wax, shellac, or 
glyptal, a disk of brass with a rod extending through it, to act as a 
terminal for a high-voltage discharge from a spark coil. The base 
plate may serve as the other terminal. This discharge is useful 

for testing the vacuum. 

A very useful device, not absolutely 
necessary, is a sylphon about 2 in. long 
soldered to the lower side of the disk. 
This is then firmly sealed to the top of the 
jar. The rod which acts as the high- 
tension terminal is made quite long, and 
equipped on its lower end with a glass 
extension. The rod may then be flexed 
so as to explore, with the glass end, a 
considerable area inside the bell jar. The 
writer has used this device for steering a 
glass shield in and out between the heat- 
ing coil and the surface to be coated. 

Instead of a bell jar, a large cylinder 
of metal is sometimes used, with a heavy 
glass plate sealed on top. This plate may also be of metal, in 
which case one or more windows about 2 in. in diameter should 

be put in. 

A stand S (Fig. 1) of convenient size is used inside the bell 
jar. On this may be mounted the mirror to be coated, suspended 
face down on a thin sheet of metal cut to size and shape. 

The diffusion pump, preferably a three-stage type with a 
cooling trap T built into its upper end, as shown in Fig. 1, 
should be equipped at its upper end with a %-in. flange to be 
sealed on the lower side of the base plate. It is absolutely 
essential that this pump be held rigidly. A convenient method is 
to make the base plate the top of a table, the four legs being 
ordinary water pipe about 1 in. in diameter. Large flanges 
fitted to the tops of these pipes may then be screwed directly 
to the base plate. The diffusion pump is then clamped in place 
to the legs with large laboratory clamps, and sealed to the base 
plate. 




FIG. 2. 



APP. V] PREPARATION OF MIRROR SURFACES 433 

An ordinary Hyvac pump will serve as a fore pump. In case 
a diffusion pump cannot be obtained, one or more charcoal 
traps may be used to aid in the evacuation of the jar. This 
process is, however, slow and tedious, especially since it is 
sometimes necessary to make several trials for a desired coat. 

The trap T in the top of the diffusion pump must be filled 
with a cooling solution or liquid air, to prevent mercury vapor 
from rising into the bell jar and contaminating the metal deposit. 
A satisfactory cooling solution is made by packing the trap with 
dry ice and slowly pouring ordinary commercial acetone over it. 
A small stopcock may be sealed to the upper part of the pump 
just below the flange to admit air to the jar. 

The heating coil should be of tungsten for evaporating most 
metals. For a few with low boiling points, such as antimony, it 
may be of nickel. A suitable diameter for tungsten wire is 
30 mils. It may be wound while red-hot into a helix to be 
mounted horizontally. The winding can be done on a steel rod 
about % in. in diameter or slightly larger. Some have a prefer- 
ence for a cone-shaped helix to be mounted vertically, acting 
as a sort of pot into which the metal is placed. The exact form 
of the heating coil should be dictated by practical considerations 
and experience, as its form is not important for ordinary mirror 
coating. 

There should be a large rheostat used in series with the heating 
coil, to control the current, and fuses inserted in the circuit for 
safety. The heating current may be 110 volts alternating 
current. 

After the coil is made and clamped in place, it should be 
preheated in a high vacuum to get rid of the oxide on its surface. 
The sylphon attachment mentioned above is useful in this 
operation, as it eliminates opening the Hbell jar and pumping 
down again after loading the coil with metal. Always after the 
coil is heated, sufficient time should elapse for it to cool before 
admitting air, so that oxidation is avoided. 

Aluminum is by far the best metal for mirror surfaces. It 
should be as pure as possible. Pure aluminum may be procured 
in pellet form which can be conveniently spaded into the coil. 
Some aluminum contains copper, which may be dissolved out 
with nitric acid. Some workers use pure aluminum wire which 
is fastened in small lengths to turns of the tungsten helix. 



434 LIGHT: PRINCIPLES AND EXPERIMENTS 

To render the apparatus airtight, the bell jar should be put in 
place and moved slightly to grind out any particles of dust 
which might adhere to its lower flange. Then, before pumping 
is started, the edge should be sealed with plasticine. A special 
plasticine in which apiezon oil is used is excellent, as the oil has a 
very low vapor pressure. For most work, however, ordinary 
plasticine will serve. Hot paraffine wax may also be 
used. 

The high-tension test coil for testing the vacuum may be an 
ordinary )^-kva. transformer, with a rating of about 10,000 volts 
across the secondary. A satisfactory vacuum for evaporation 
is reached when no discharge is possible between the upper 
terminal and the base plate. When this vacuum is attained, 
turn on the heating current slowly, making sure the cooling 
water is flowing through the terminals. 

A suitable deposit is a matter of experience. 

3. Deposition by Cathodic Sputtering. The same bell jar as 
for evaporation may be used, except that to the electrode at the 
top of the bell jar should be attached firmly a disk of the metal 
to be sputtered. The disk should be slightly larger than the 
metal to be sputtered. This disk is to be the cathode of a high- 
tension discharge. A suitable source is the ^2-kva. transformer 
mentioned in Sec. 2 above, but in this case the second terminal 
in the apparatus must either be a small point shielded from the 
mirror surface or it must be removed as far as possible from the 
mirror surface. This may be accomplished by having a side 
tube of about 1 in. diameter sealed to the tube connecting the 
base plate and the diffusion pump, just below the flange. This 
side tube should be about 6 in. long and have an aluminum 
electrode sealed into the end of it. 

Sputtering must be done with a higher pressure than evapora- 
tion. This pressure may be calibrated roughly by the width 
of the cathode dark space, which grows as the pressure drops. 
A dark space of about 3 cm. indicates a satisfactory pressure. 
The mirror to be coated should be mounted face up, below the 
cathode, and just inside the cathode dark space. 

Aluminum does not sputter well. Silver may be used suc- 
cessfully, and is by far the best metal if this method of deposition 
is to be used. The exact amount of deposit for a suitable mirror 
is a matter of experience. Sputtering is found to be most 



Apr. VI] MAKING CROSS HAIRS 435 

successful in an atmosphere of some rare gas, such as helium, 
argon, or neon. 

Additional details of cathodic sputtering and evaporation may 
be found in the following articles: 

"Making of Mirrors by the Deposition of Metal on Glass," 
Bureau of Standards Circular 389, 1931 (chemical deposition and 
cathodic sputtering). 

JONES, E. G., and E. W. FOSTER, "Production of Silver 
Mirrors by Kathodic Sputtering," Journal of Scientific Instru- 
ments (London), 13, 216, 1936. 

WILLIAMS, R. C., and G. B. SABINE, "Evaporated Films for 
Large Mirrors," Astrophysical Journal, 77, 316, 1933. 

STRONG, JOHN, " Aluminizing of Large Telescope Mirrors," 
Astrophysical Journal, 83, 401, 1936 (evaporation). 

APPENDIX VI 
MAKING CROSS HAIRS 

One of the time-honored methods of making cross hairs is to 
fasten spider-web strands, silk fibers, or similar filaments on a 
metal holder with fast-drying cement, 
such as shellac. Another method, 
especially useful where two or more 
lines close together and parallel are 
desired, is to rule them with a diamond 
on a glass disk. Both of these meth- 
ods involve a considerable amount of - , 

rio. 1. 

technical skill, and the second requires 

apparatus which is often beyond the means of the laboratory. 
Recently it has been discovered that filaments spun of some 
quick-drying cement make excellent crdss hairs. 

A small drop of fresh Duco is squeezed on the end of a match 
stick or pencil, touched immediately to one side of the holder, and 
drawn into a fine filament which is laid across the other side 
of the holder so that it sticks there. Since this often results in a 
filament which does not have a uniform diameter in the field of 
view of the telescope, the following modification is recommended: 
Make out of wood or metal a small frame as shown in Fig. 1, 
in which the holder is held securely, with the surface on which 
the cross hairs are to be mounted slightly above the upper surface 




436 LIGHT: PRINCIPLES AND EXPERIMENTS 

of the frame. The fresh Duco is touched at A, drawn quickly 
to a filament which is lowered to points B and C on the holder, 
and fastened at D. Then a small drop of Duco is laid on the 
filament at B and C to anchor it securely. The Duco must be 
quite fresh or it will not spin properly. With a little practice, 
extremely fine cross hairs can be made in this manner. It is 
recommended that the entire operation be carried on under a 
hand magnifier or equivalent lens mounted in position above 
the frame. 

APPENDIX VII 
STANDARD SOURCES FOR COLORIMETRY 1 

It is recommended that three illuminants A, B, and C as 
described below, be adopted as standards for the general col- 
orimetry of materials. 

A. A gas-filled lamp operated at a color temperature of 
2848K. 

B. The same lamp used in combination with a filter consisting 
of a layer, 1 cm. thick of each of two solutions B\ and J5 2 , con- 
tained in a double cell made of colorless optical glass. The 
solutions are to be made up as follows: 

Solution B\: 

Copper sulphate (CuSO 4 .5H 2 O) 2. 452 grams 

Mannite [C 6 H 8 (OH) fi ] 2. 452 grams 

Pyridine (C 5 H 5 N) 30.0 c.c. 

Water (distilled) to make 1000.0 c.c. 

Solution B 2 : 

Cobalt ammonium sulphate 

[CoSO 4 .(NH 4 ) 2 SO 4 .6H 2 O] 21 . 71 grams 

Copper sulphate (CuSO 4 .5H 2 O) 16.11 grams 

Sulphuric acid (sp. gr. 1.835) 10.0 c.c. 

Water (distilled) to make 1000.0 c.c. 

C. The same lamp used in combination with a filter consisting 
of a layer, 1 cm. thick of each of two solutions C\ and C 2 , contained 
in a double cell made of colorless optical glass. The solutions 
are to be made up as follows : 

1 Taken from T. SMITH and J. GUILD, "The C.I.E. Colorimetric Standards 
and Their Use," Transactions of the Optical Society, London, 33, 73, 1931-1932. 



APP. VII] STANDARD SOURCES FOR COLORIMETRY 



437 



Solution C\\ 

Copper sulphate (CuSO 4 .5H 2 O) 3. 412 grams 

Mannite [C 6 H 8 (OH) 6 ] 3. 412 grams 

Pyridine (CjH 6 N) 30. c.c. 

Water (distilled) to make 1000. c.c. 

Solution C-i\ 

Cobalt ammonium sulphate 

[CoSO 4 .(NH 4 ) 2 S0 4 .6H 2 0] 30. 580 grams 

Copper sulphate (CuSO 4 .5H 2 O) 22 . 520 grams 

Sulphuric acid (sp. gr. 1.835) 10.0 c.c. 

Water (distilled) to make 1000. c.c. 

It is also recommended that the following spectral-energy 
distribution values for each of these illuminants shall be used in 
computation of colorimetric quantities from spectrophotometric 
measurements . 

Source A. The spectral distribution of energy from this source 
may be taken for all colorimetric purposes to be that of a black 
body at a temperature of 2848K. The value assumed for 
Planck's constant c$ is 14,350 micron-deg. 

SPECTRAL DISTRIBUTION OF ENERGY; SOURCES B AND C 



Wave- 
length, 
angstroms 


Relative energy 


Wave- 
length, 
angstroms 


Relative energy 


B 


C 


B 


C 


3700 


15.2 


21.6 


5500 


101.0 


105.2 


3800 


22.4 


33.0 


5600 102.8 


105.3 


3900 


31.3 


47.4 


5700 


102.6 


102.3 


4000 


41.3 


63.3 


5800 101.0 


97.8 


4100 


52.1 


80.6 


5900 99.2 


93.2 


4200 


(53.2 


98.1 


6000 98.0 


89.7 


4300 
4400 


73.1 
80.8 


112.4 
121.5 


61QO 98.5 
6200 99.7 


88.4 
88.1 


4500 


85.4 


124.0 


6300 101.0 


88.0 


4600 


88.3 


123.1 


6400 


102.2 


87.8 


4700 


92.0 


123.8 


6500 103.9 


88.2 


4800 


95.2 


123.9 


6600 105.0 


87.9 


4900 


96.5 


120.7 


6700 


104.9 


86.3 


5000 


94.2 


112.1 


6800 


103.9 


84.0 


5100 


90.7 


102.3 


6900 


101.6 


80.2 


5200 


89.5 


96.9 


7000 


99.1 


76.3 


5300 


92.2 


98.0 


7100 


96.2 


72.4 


5400 


96.9 


102.1 


7200 


92.9 


68.3 



438 LIGHT: PRINCIPLES AND EXPERIMENTS 

Sources B and C. The spectral distributions of energies 
for these sources, as computed from the spectrophotometric 
measurements of the transmission of the filters made by Messrs. 
Davis and Gibson of the Bureau of Standards, are tabulated 
on p. 437. 

APPENDIX VIII 
THE FRESNEL INTEGRALS 

In Sec. 12-6 a vector-polygon method has been described by 

which the amplitude of the distribution due to any part of a 

light wave may be evaluated. When 
the separate elements of the disturb- 
ance are taken small enough, the 
vectors representing them become a 
curve which for an unobstructed wave 
front is the Cornu spiral (Fig. 12-9) 
X of which a drawing to scale is included 
in this appendix. Cornu originally 
constructed this spiral by plotting 

the values of Fresnel's integrals, which may be derived in the 

following manner: 

Consider such a curve (Fig. 1) representing the summa- 

tion of a number of elements of amplitude of a wave disturbance. 

Let x and y be the coordinates of an element of disturbance dS. 

Then the angle <j> between the z-axis and the tangent to the 

curve is the phase of the element dS. We may write 

, dx . dy 

cos * - , mn 4- = . 




x = / cos dS, y = J sin <> dS. (1) 

It is now necessary to evaluate <f> and dS in terms of an actual 
wave front. For this purpose we may consider a cylindrical 
wave front W originating at a line source L (Fig. 2), perpendicular 
to the paper. It is required to find the intensity at a point 
on the screen. By the cosine law, and substituting for 9 its 
approximate value s/a, 

c* = (a + 6) 2 -fa 2 - 2a(a + 6) cos -, 

(Ji 



APP. VIII] 



THE FRESNEL INTEGRALS 



439 



or 



or 



c 2 = 6 2 H s 2 , approximately, 



a + 6 
2a6 






This is sufficiently accurate when 8 is small. 

The difference of phase be- 
tween the disturbance at due 
to its pole and that due to dW 
is measured by (c &)/X, so 
that, if the first is given by sin 
2irt/T, the second is given by 



sin 27r( ~ 




FIG. 2. 



D 



(3) 



and the entire disturbance due to all the elements by 

= Jsin 2 

The integral is taken between limits appropriate to the particular 
case. 

The amplitude contributed to O by an element of the wave 
front dW is proportional to its area, inversely proportional to 
the distance c from dW, and depends also on the obliquity of the 
wave front. If we neglect these considerations and assume 
merely that the amplitude due to any element is proportional 
to the length of the element, then we may identify dW in eq. 3 
as dS in eqs. 1. Similarly the phase angle <f> may be related to 



the path difference (c 6), for </> = (c 6), so that by eq. 2 

A 

(a + *>X 2 



Substituting v 2 for (a + b)s 2 /a6X, we may now write the expres- 
sion for the intensity / due to the wave front in terms of the x and 
y coordinates given in eqs. 1: 



;[(/<-? 




sin 



^ 

(4) 



440 



LIGHT: PRINCIPLES AND EXPERIMENTS 



The integrals in eq. 4 are known as Fresnel'i* integrals. They 
have been evaluated by Gilbert, 1 and appear in the following 
table. 

TABLE OF FREBNEL'B INTEGRALS 



t'l 


f 91 ***j 
Jo ow-2-* 


/* . irtf* 

Jo ""-r* 


Vl 


/* irt; J 
J -ar* 


r . irt> 
sin -yA, 


0.0 


0.0000 


o.oooo 


2.6 


0.3889 


0.5500 


0.1 


0.1000 


0.0005 


2.7 


0.3926 


0.4529 


0.2 


0.1999 


0.0042 


2.8 


0.4675 


0.3915 


0.3 


0.2994 


0.0141 


2.9 


0.5624 


0.4102 


0.4 


0.3975 


0.0334 


3.0 


0.6057 


0.4963 


0.5 


0.4923 


0.0647 


3.1 


0.5616 


0.5818 


0.6 


0.5811 


0.1105 


32 


0.4663 


0.5933 


0.7 


0.6597 


0.1721 


3.3 


4057 


0.5193 


0.8 


0.7230 


0.2493 


3.4 


0.4385 


0.4297 


0.9 


0.7648 


0.3398 


3.5 


0.5326 


0.4153 


1.0 


0.7799 


0.4383 


3.6 


0.5880 


0.4923 


1.1 


0.7638 


0.5365 


37 


0.5419 , 0.5760 


1.2 


0.7154 


0.6234 


3 8 


0.4481 


0.5656 


1.3 


0.6386 


0.6863 


3.9 


0.4223 


0.4752 


1.4 


0.5431 0.7135 


4.0 0.4984 


0.4205 




i 


1 




1.5 


0.4453 


0.6975 


4. 1 


0.5737 


0.4758 


1.6 


0.3655 


0.6383 


4.2 


5417 


0.5632 


1.7 


3238 


0.5492 


4.3 


0.4494 


0.5540 


1.8 


0.3337 


0.4509 


4 4 


0.4383 


0.4623 


1.9 


0.3945 


0.3734 


4.5 


0.5258 


0.4342 


2.0 


0.4883 


0.3434 


4.6 


0.5672 


0.5162 


2.1 


0.5814 


0.3743 


4.7 


0.4914 


0.5669 


2.2 


0.6362 


0.4556 


4.8 


0.4338 


0.4968 


2.3 


0.6268 


0.5525 


4.9 


0.5002 


0.4351 


2.4 


0.5550 


0.6197 


5.0 


0.5636 


0.4992 


2.5 


0.4574 


0.6192 


00 


0.5000 


0.5000 




J 









From these values the Cornu spiral shown in Fig. 3 was plotted 
on a large scale and reduced photographically. 

The assumptions made in this derivation, that is small, and 
that dW is proportional to the length of the element , are tanta- 
mount to the assumption that the effective portion of the cylin- 
drical wave front is really confined to a very small area about 
the pole of any point O under consideration. 

1 GILBERT, PHILIPPE, Acad6mie Royale de Belgique, 31, 1, 1863. Correc- 
tions have been made to his values of the cosine term for v\ equal to 0.1 
and 1.8. 



\ 




<N 



d 
'3 



a 



Ji 

** 

O 
+ 

(3 

1 



C9 

<41 

O 





& 



441 



TABLES OF USEFUL DATA 



TABLE I. USEFUL WAVE-LENGTHS 

The wave-lengths listed are principally those of lines which may be 
obtained with discharge tubes of helium, hydrogen, neon, mercury; with 
the mercury arc, or in the spectrum of the sun. Only the stronger lines 
due to these sources are listed. There are a few, such as the cadmium lines, 
which may be obtained with an ordinary mercury arc containing an aftialgam 
of mercury and the other metals desired. The values given are in angstroms 
(1 angstrom = 10~ 8 cm.). 

In any particular source there may appear lines fainter than those listed, 
or lines due to impurities. If the wave-lengths of such lines are measured, 
they may usually be identified by consulting H. Kayser, "Tabelle der 
Hauptlinien der Linienspektra aller Elemente," published by Julius Springer, 
or Twyman and Smith, "Wave-length Tables for Spectrum Analysis," 
published by Adam Hilger, Ltd. An extensive table of wave-lengths is also 
included in the more recent editions of the "Handbook of Chemistry and 
Physics," published by the Chemical Rubber Publishing Co. 



Hydrogen 
6562.8 H 
4861 . 3 H/j 
4340.5 
4101 . 7 
3970.1 
3889.0 



H 7 



H 
H f 



4339.2 
4347.5 
4358? 3 

4077.8 
4046.8 
4046.6 | 

Fraunhofer Lines 



Helium 
7065.2 
6678.1 
5875.6 



5047. 
5015. 



4921 . 9 
4713.1 
4471.5 
4437.5 
4387.9 
4143.8 
4120.8 
4026.2 
3964.7 
3888.6 

Neon 
6929.5 
6717.0 
6678.3 
6599.0 
6532.9 

i If the solar spectrum is 
seen owing to a blend of Fe 



B 
C 
D 2 
D, 
E 
F 
G' 
H 
K 


6870. 
6562. 
5895. 
5889. 
5270. 
4861. 
4340. 
3968. 
3933. 



8 
93 
96 
1 
3 
5 
5 
7 


2 
H 
Na 
Na 
FeCa 
H 
H 1 
Call 
Call 



6500.5 
6402 2 
6383.0 
6334.4 
6304.8 
6266.5 
6217.3 
6163.6 
6143.1 
6096.2 
6074.3 
6030.0 
5975.5 
5944.8 
5881.9 
5852.5 
5820.2 
5764.4 
5400.6 
5341.1 
5330.8 

Mercury 
^6234.3 
6123.5 
6072.6 
5790. 
5769.60 
5460. 
4916.0 



used, with small dispersion, a wide absorption line also will be 
and Ca lines, with a mean wave-length of 4307.8 angstroms. 
443 



Miscellaneous 




6707.9 


Li 


6438.5 


Cd 


6103.6 


Li 


5895.93 


Na 


5889.96 


Na 


5535.5 


Ba 


5085.8 


Cd 


4799.9 


Cd 


4678.2 


Cd 


3968.5 


Call 


*3933 . 7 


Call 



444 



LIGHT: PRINCIPLES AND EXPERIMENTS 



TABLE II. INDICES OF REFRACTION OF SOME COMMON SUBSTANCES 
a. Glasses and Optically Isotropic Substances. In specifying glass, the 
manufacturer usually gives n/>, the index for the sodium lines, and also the 
value of v = (nD l)/(np nc), the indices for several other lines of 
common sources, and the differences between these and a number of other 
lines. From these data a dispersion curve may be drawn. In the following 
table of representative glasses, the indices are given at intervals of 600 
angstroms from 4000 to 7500 angstroms, from which the index for any other 
line may be obtained with an accuracy sufficient for the experiments and 
problems in this book. For more precise information for a given specimen 
of glass the manufacturer should be consulted. Detailed information con- 
cerning many glasses is to be found in the International Critical Tables. 



Wave-length, 
angstroms 


Light 
crown 


Dense 
crown 


Light 
flint 


Dense 
flint 


Heavy 
flint 


Fused 
quartz 


Fluor- 
ite 


4000 


1 . 5238 


1 . 5854 


1 . 5932 


1.6912 


1.8059 


1 . 4699 


1 . 4421 


4500 


1 . 5180 


1.5801 


1 . 5853 


1 . 6771 


1 . 7843 


1 . 4655 


1.4390 


5000 


1.5139 


1 . 5751 


1.5796 


1 . 6670 


1 . 7706 


1 . 4624 


1 . 4366 


5500 


1 . 5108 


1.5732 


1.5757 


1 . 6591 


1.7611 


1 . 4599 


1.4350 


6000 


1.5085 


1 . 5679 


1 . 5728 


1 . 6542 


1 . 7539 


1 . 4581 


1 . 4336 


6500 


1 . 5067 


1 . 5651 


1.5703 


1 . 6503 


1 . 7485 


1 . 4566 


1 . 4324 


7000 


1 . 5051 


1 . 5640 


1 . 5684 


1 . 6473 


1 . 7435 


1 . 4553 


1.4318 


7500 


1.5040 


1 . 5625 


1 . 5668 


1 . 6450 


1 . 7389 


1 . 4542 


1.4311 



b. Liquids. 



Substance 


Temperature, 
C. 


Index with 
respect to 
air for 
D-lines 


Water 


20 


1.3330 


Acetone 


20 


1.359 


Ammonia (d 0.615) liquid 
Benzene ... 


16.5 
20 


1.325 
1.501 


Bromine 


20 


1 . 654 


Carbon disulphide 


18 


1 . 6255 


Carbon dioxido 


15 


1.195 


Chloroform 


20 


1.446 


Ethyl ether 


22 


1.351 


Ethyl alcohol 


20 


1.3605 


Glvcerin ... 


20 


1.474 


Methyl alcohol 
Toluene 


20 
20 


1.329 
1.495 









c. The index of refraction of air at 0C. and 760 mm. Hg pressure with 
respect to a vacuum is 1.0002926. 



TABLES OF USEFUL DATA 



445 



d. Some Uniaxial Crystal*. 





Index for 1 


,he D lines 


Substance 


Ordinary ray 


Extraordinary 
ray 


Calcite 


1.658 


1 486 


Ice 


1.3091 


1 3104 


Quartz 


1 . 54424 


1 . 55335 


Tourmaline 


1 669 


1 638 


Sodium nitrate 


1 . 5874 


1 3361 


Zircon 


1 . 923 


1.968 



TABLE III. REFLECTING POWERS OF SOME METALS 
Since the measured reflecting power varies widely with the origin of the 
surface and its age, these factors should be taken into account in using the 
figures given below. The values given for silver, aluminum, and gold are 
compiled from graphical data in an article on the evaporating process by 
John Strong in Astrophysical Journal, 83, 401, 1936, and are for freshly 
evaporated opaque coatings. Experience shows that for the visible region 
the reflecting power of silver diminishes between 15 and 20 per cent in two 
or three weeks' time. The values for platinum, copper, steel, monel, and 
speculum are for polished massive metals. In general these have less 
reflecting power than the evaporated coats of the same metals. Additional 
data may be found in the International Critical Tables, 



x, 


Percentage reflection of normally incident light 


















angstroms 


Silver 


Alum- 


Plati- 


Spec- 

* 


Steel 


Monel 


Gold 


Cop- 






inum 


num 


ulum 








per 


2500 


34* 


80 


33 


29 


33 




39* 


26 


3000 


9* 


85 


39 


40 


37 


. . 


32* 


25 


3500 


74 


87 


43 


51 


44 


. . 


28 


27 


4000 


89 


89 


48 


55 


50 


, . 


28 


31 


4500 


93 


90 


55 


60 


54 


56 


33 


37 


5000 


95 


90 


58 


63 


55 


58 


47 


44 


5500 


95 


89 


61 


64 


55 


59 


74 


48 


6000 


95 


89 


64 


64 


55 


60 


84 


72 


6500 


96 


87 


66 


65 


56 


62 


89 


80 


7000 


97 


87 


69 


69 


57 


64 


92 


83 


7500 


97 


86 


69 


69 


58 


66 


93 


86 


8000 


97 


85 


70 


70 


58 


67 


95 


89 



* From other sources than those indicated above. 



446 



LIGHT: PRINCIPLES AND EXPERIMENTS 



TABLE IV. FOUR-PLACE LOGARITHMIC TABLES 



Logarithms 


Mean Differences 







1 


2 


3 


4 


5 


6 


7 


8 


9 


123 


456 


789 


10 
11 
12 
13 
14 


0000 
0414 
0792 
1139 
1461 


0043 
0453 
0828 
1173 
1492 


0086 
0492 
0864 
1206 
1523 


0128 
0531 
0899 
1239 
1553 


0170 
0569 
0934 
1271 
1584 


0212 
0607 
0969 
1303 
1614 


0253 
0645 
1004 
1335 
1644 


0294 
0682 
1038 
1367 
1673 


0334 
0719 
1072 
1399 
1703 


0374 
0755 
1106 
1430 
1732 


4 8 12 
4 8 11 
3 7 10 
3 6 10 
369 


17 21 25 
15 19 23 
14 17 21 
13 16 19 
12 15 18 


29 33 37 
26 30 34 
24 28 31 
23 26 29 
21 24 27 


15 
16 
17 
18 
19 


1761 
2041 
2304 
2553 

2788 


1790 
2068 
2330 
2577 
2810 


1818 
2095 
2355 
2601 
2833 


1847 

2122 
2380 
2625 
2856 


1875 
2148 
2405 
2648 

2878 


1903 
2175 
2430 
2672 
2900 


1931 
2201 
2455 
2695 
2923 


1959 
2227 
2480 
2718 
2945 


1987 
2253 
2504 
2742 
2967 


2014 
2279 
2529 
2765 
2989 


368 
358 
257 
257 
247 


11 14 17 
11 13 16 
10 12 15 
9 12 14 
9 11 13 


20 22 25 
18 21 24 
17 20 22 
16 19 21 
16 18 20 


20 
21 
22 
23 
24 


3010 
3222 
3424 
3617 
3802 


3032 
3243 
3444 
3636 
3820 


3054 
3263 
3464 
3655 
3838 


3075 
3284 
3483 
3674 
3856 


3096 
3304 
3502 
3692 
3874 


3118 
3324 
3522 
3711 
3892 


3139 
3345 
3541 
3729 
3909 


3160 
3365 
3560 
3747 
3927 


3181 
3385 
3579 
3766 
3945 


3201 
3404 
3598 
3784 
3962 


246 
246 
246 
246 
245 


8 11 13 
8 10 12 
8 10 12 
7 9 11 
7 9 11 


15 17 19 
14 16 18 
14 15 17 
13 15 17 
12 14 16 


25 
26 
27 
28 
29 


3979 
4150 
4314 
4472 
4624 


3997 
4166 
4330 
4487 
4639 


4014 
4183 
4346 
4502 
4654 


4031 
4200 
4362 
4518 
4669 


4048 
4216 
4378 
4533 
4683 


4065 
4232 
4393 
4548 
4698 


4082 
4249 
4409 
4564 
4713 


4099 
4265 
4425 
4579 

4728 


4116 
4281 
4440 
4594 
4742 


4133 
4298 
4456 
4609 

4757 


235 
235 
235 
235 
1 3 4 


7 9 10 
7 8 10 
689 
689 
679 


12 14 15 
11 13 15 
11 13 14 
11 12 14 
10 12 13 


30 
31 
32 
33 
34 


4771 
4914 
5051 
5185 
5315 


4786 
4928 
5065 
5198 

5328 


4800 
4942 
5079 
5211 
5340 


4814 
4955 
5092 
5224 
5353 


4829 
4969 
5105 
5237 
5366 


4843 
4983 
5119 
5250 
5378 


4857 
4997 
5132 
5263 
5391 


4871 
5011 
5145 
5276 
5403 


4886 
5024 
5159 
5289 
5416 


4900 
5038 
5172 
5302 
5428 


1 3 4 
1 3 4 
134 
134 
1 3 4 


679 
678 
578 
568 
568 


10 11 13 
10 11 12 
9 11 12 
9 10 12 
9 10 11 


36 
36 
37 
38 
39 


5441 
5563 
5682 
5798 
5911 


5453 
5575 
5694 
5809 
5922 


5465 
5587 
5705 
5821 
5933 


5478 
5599 
5717 
5832 
5944 


5490 
5611 
5729 
5843 
5955 


5502 
5623 
5740 
5855 
5966 


5514 
5635 
5752 

5866 
5977 


5527 
5647 
5763 

5877 
5988 


5539 
5658 
5775 
5888 
5999 


5551 
5670 
5786 
5899 
6010 


1 2 4 
1 2 4 
1 2 3 
1 2 3 
1 2 3 


567 
567 
567 
567 
457 


9 10 11 
8 10 11 
8 9 10 
8 9 10 
8 9 10 


40 
41 
42 
43 
44 


6021 
6128 
6232 
6335 
6435 


6031 
6138 
6243 
6345 
6444 


6042 
6149 
6253 
6355 
6454 


6053 
6160 
6263 
6365 
6464 


6064 
6170 
6274 
6375 
6474 


6075 
6180 
6284 
6385 
6484 


6085 
6191 
6294 
6395 
6493 


6096 
6201 
6304 
6405 
6503 


6107 
6212 
6314 
6415 
6513 


6117 
6222 
6325 
6425 
6522 


1 2 3 
123 
1 2 3 
123 
123 


456 
456 
456 
456 
456 


8 9 10 
789 
789 
789 
789 


46 
46 
47 
48 
49 


6532 
6628 
6721 
6812 
6902 


6542 
6637 
6730 
6821 
6911 


6551 
6646 
6739 
6830 
6920 


6561 
6656 
6749 
6839 
6928 


6571 
6665 
6758 
6848 
6937 


6580 
6675 
6767 
6857 
6946 


6590 
6684 
6776 
6866 
6955 


(>599 
6693 
6785 
6875 
6964 


6609 
6702 
6794 
6884 
6972 


6618 
6712 
6803 
6893 
6981 


1 2 3 
123 
123 
123 
123 


456 
456 
455 
445 
445 


789 
778 
678 
678 
678 


50 
61 
62 
53 
54 


6990 
7076 
7160 
7243 
7324 


6998 
7084 
7168 
7251 
7332 


7007 
7093 
7177 
7259 
7340 


7016 
7101 
7185 
7267 
7348 


7024 
7110 
7193 
7275 
7356 


7033 
7118 
7202 
7284 
7364 


7042 
7126 
7210 
7292 
7372 


7050 
7135 
7218 
7300 
7380 


7059 
7143 
7226 
7308 
7388 


7067 
7152 
7235 
7316 
7396 


123 
1 2 3 
122 
1 2 2 
1 2 2 


345 
345 
345 
345 
345 


678 
678 
677 
667 
667 







1 


2 


3 


4 


5 


6 


7 


8 


9 


123 











TABLES OF USEFUL DATA 



447 



TABLE IV. FOUR-PLACE LOGARITHMIC TABLES. (Continued] 



Logarithms 


Mean Differences 







1 


2 


3 


4 


6 


6 


7 


8 


9 


123 


456 


789 


56 
56 
67 
58 
69 


7404 
7482 
7559 
7634 
7709 


7412 
7490 
7566 
7642 
7716 


7419 
7497 
7574 
7649 
7723 


7427 
7505 

7582 
7657 
7731 


7435 
7513 
7589 
7664 
7738 


7443 
7520 
7597 
7672 
7745 


7451 
7528 
7604 
7679 
7752 


7459 
7536 
7612 
7686 
7760 


7466 
7543 
7619 
7694 
7767 


7474 
7551 
7627 
7701 

7774 


1 2 2 
1 2 2 
122 
1 1 2 
1 1 2 


345 
345 
345 
344 
344 


507 
567 
567 
567 
567 


60 
61 
62 
63 
64 


7782 
7853 
7924 
7993 
8062 


7789 
7860 
7931 
8000 
8069 


7796 
7868 
7938 
8007 
8075 


7803 

7875 
7945 
8014 
8082 


7810 
7882 
7952 
8021 
8089 


7818 
7889 
7959 
8028 
8096 


7825 
7896 
7966 
8035 
8102 


7832 
7903 
7973 
8041 
8109 


7839 
7910 
7980 
8048 
8116 


7846 
7917 
7987 
8055 
8122 


1 1 2 
1 1 2 
1 1 2 
1 1 2 
1 1 2 


344 
344 
334 
334 
334 


566 
566 
566 
556 
556 


65 
66 
67 
68 
69 


8129 
8195 
8261 
8325 
8388 


8136 
8202 
8267 
8331 
8395 


8142 
8209 
8274 
8338 
8401 


8149 
8215 
8280 
8344 
8407 


8156 
8222 
8287 
8351 
8414 


8162 
8228 
8293 
8357 
8420 


8169 
8235 
8299 
8363 
8426 


8176 
8241 
8306 
8370 
8432 


8182 
8248 
8312 
8376 
8439 


8189 
8254 
8319 
8382 
8445 


1 1 2 
112 
1 1 2 
1 1 2 
1 1 2 


334 

a 3 4 

334 
334 
234 


556 
556 
556 
456 
456 


70 
71 
72 
73 
74 


8451 
8513 
8573 
8633 
8692 


8457 
8519 
8579 
8639 
8698 


8463 
8525 
8585 
8645 
8704 


8470 
8531 
8591 
8651 
8710 


8476 
8537 
8597 
8657 
8716 


8482 
8543 
8603 
8663 
8722 


8488 
8549 
8609 
8669 

8727 


8494 
8555 
8615 
8675 
8733 


8500 
8561 
8621 
8681 
8739 


8506 
8567 
8627 
8686 
8745 


1 1 2 
1 1 2 
112 
112 
1 1 2 


234 
234 
234 
234 
234 


456 
455 
455 
455 
455 


75 
76 
77 
78 
79 


8751 
8808 
8865 
8921 
8976 


8756 
8814 
8871 
8927 
8982 


8762 
8820 
8876 
8932 
8987 


8768 
8825 
8882 
8938 
8993 


8774 
8831 
8887 
8943 
8998 


8779 
8837 
8893 
8949 
9004 


8785 
8842 
8899 
8954 
9009 


8791 
8848 
8904 
8960 
9015 


8797 
8854 
8910 
8965 
9020 


8802 
8859 
8915 
8971 
9025 


1 1 2 
1 1 2 
1 1 2 
1 1 2 
112 


233 
233 
233 
2 3 3 
2 3 3 


455 

455 
445 
445 

445 


80 
81 
82 
83 
84 


9031 
9085 
9138 
9191 
9243 


9036 
9090 
9143 
9196 
9248 


9042 
9096 
9149 
9201 
9253 


9047 
9101 
9154 
9206 
9258 


9053 
9106 
9159 
9212 
9263 


9058 
9112 
9165 
9217 
9269 


9063 
9117 
9170 
9222 
9274 


9069 
9122 
9175 
9227 
9279 


9074 
9128 
9180 
9232 
9284 


9079 
9133 
9186 
9238 
9289 


1 1 2 
1 1 2 
1 1 2 
1 1 2 
1 1 2 


233 
2 3 3 
233 
233 
2 3 3 


445 
445 
445 
445 
445 


85 
86 
87 
88 
89 


9294 
9345 
9395 
9445 
9494 


9299 
9350 
9400 
9450 
9499 


9304 
9355 
9405 
9455 
9504 


9309 
9360 
9410 
9460 
9509 


9315 
9365 
9415 
9465 
9513 


9320 
9370 
9420 
9469 
9518 


9325 
9375 
9425 
9474 
9523 


9330 
9380 
9430 
9479 
9528 


9335 
9385 
9435 
9484 
9533 


9340 
9390 
9440 
9489 
9538 


1 1 2 
1 1 2 
1 1 
1 1 
1 1 


233 
233 
223 
223 
223 


445 
445 
344 
344 
344 


90 
91 
92 
93 
94 


9542 
9590 
9638 
9685 
9731 


9547 
9595 
9643 
9689 
9736 


9552 
9600 
9647 
9694 
9741 


9557 
9605 
9652 
9699 
9745 


9562 
9609 
9657 
9703 
9750 


9566 
9614 
9661 
9708 
9754 


9571 
9619 
9666 
9713 
9759 


9576 
9624 
9671 
9717 
7763 


9581 
9628 
9675 
9722 
9768 


9586 
9633 
9680 
9727 
9773 


1 1 
1 1 
1 1 
1 1 
1 1 


223 
223 
223 
223 
223 


344 
344 
344 
344 
344 


96 
96 
97 
98 
99 


9777 
9823 
9868 
9912 
9956 


9782 
9827 
9872 
9917 
9961 


9786 
9832 
9877 
9921 
9965 


9791 
9836 
9881 
9926 
9969 


9795 
9841 
9886 
9930 
9974 


9800 
9845 
9890 
9934 
9978 


9805 
9850 
9894 
9939 
9983 

6 


9809 
9854 
9899 
9943 
9987 


9814 
9859 
9903 
9948 
9991 

8 


9818 
9863 
9908 
9952 
9996 


1 1 
1 1 
1 1 
1 1 
1 1 


223 
223 
223 
223 
223 


344 
344 
344 
344 
334 







1 


2 


3 


4 


5 


7 


9 


123 


456 


789 



448 



LIGHT: PRINCIPLES AND EXPERIMENTS 



TABLE V. TRIGONOMETRIC FUNCTIONS 



Natural Sines 0-45 


Mean Differences 
(Add) 


O 


0' 


10' 


20' 


30' 


40' 


60' 






















.0000 


.0029 


.0058 


.0087 


.0116 


.0145 


.0175 


89 


360 


12 14 17 


20 23 26 


1 


.0175 


.0204 


.0233 


.0262 


.0291 


.0320 


.0349 


88 


369 


12 14 17 


20 23 26 


2 


.0349 


.0378 


.0407 


.0436 


.0465 


.0494 


.0523 


87 


360 


12 14 17 


20 23 26 


3 


.0523 


.0552 


.0581 


.0610 


.0640 


.0669 


.0698 


86 


369 


12 14 17 


20 23 26 


4 


.0698 


.0727 


.0756 


.0785 


.0814 


.0843 


.0872 


86 


369 


12 14 17 


20 23 26 


5 


.0872 


.0901 


.0929 


.0958 


.0987 


.1016 


.1045 


84 


369 


12 14 17 


20 23 26 


6 


.1045 


.1074 


.1103 


.1132 


.1161 


.1190 


.1219 


83 


369 


12 14 17 


20 23 26 


7 


.1219 


.1248 


.1276 


.1305 


.1334 


.1363 


.1392 


82 


369 


12 14 17 


20 23 26 


8 


.1392 


.1421 


.1449 


.1478 


.1507 


.1536 


.1564 


81 


369 


12 14 17 


20 23 26 


9 


.1564 


.1593 


.1622 


.1650 


.1679 


.1708 


.1736 


80 


369 


12 14 17 


20 23 26 


10 


.1736 


.1765 


.1794 


.1822 


.1851 


.1880 


.1908 


79 


369 


12 14 17 


20 23 26 


11 


.1908 


.1937 


.1965 


.1994 


.2022 


.2051 


.2079 


78 


368 


11 14 17 


20 22 25 


12 


.2079 


.2108 


.2136 


.2164 


.2193 


.2221 


.2250 


77 


368 


11 14 17 


20 22 25 


13 


.2250 


.2278 


.2306 


.2334 


.2363 


.2391 


.2419 


76 


368 


11 14 17 


20 22 25 


14 


.2419 


.2447 


.2476 


.2504 


.2532 


.2560 


.2588 


76 


368 


11 14 17 


20 22 25 


16 


.2588 


.2616 


.2644 


.2672 


.2700 


.2728 


.2756 


74 


368 


11 14 17 


20 22 25 


16 


.2756 


.2784 


.2812 


.2840 


.2868 


.2896 


.2924 


73 


368 


11 14 17 


20 22 25 


17 


.2924 


.2952 


.2979 


.3007 


.3035 


.3062 


.3090 


72 


368 


11 14 17 


20 22 25 


18 


.3090 


.3118 


.3145 


.3173 


.3201 


.3228 


.3256 


71 


368 


11 14 17 


20 22 25 


19 


.3256 


.3283 


.3311 


.3338 


.3365 


.3393 


.3420 


70 


358 


11 14 16 


19 22 24 


20 


.3420 


.3448 


.3475 


.3502 


.3529 


.3557 


.3584 


69 


358 


11 14 16 


19 22 24 


21 


.3584 


.3611 


.3638 


.3665 


.3692 


.3719 


.3746 


68 


358 


11 14 16 


19 22 24 


22 


.3746 


.3773 


.3800 


.3827 


.3854 


.3881 


.3907 


67 


358 


11 14 16 


19 22 24 


23 


.3907 


.3934 


.3961 


.3987 


.4014 


.4041 


.4067 


66 


358 


11 14 16 


19 22 24 


24 


.4067 


.4094 


.4120 


.4147 


.4173 


.4200 


.4226 


66 


358 


11 14 16 


19 22 24 


26 


.4226 


.4253 


.4279 


.4305 


.4331 


.4358 


.4384 


64 


358 


10 13 16 


18 21 23 


26 


.4384 


.4410 


.4436 


.4462 


.4488 


.4514 


.4540 


63 


358 


10 13 16 


18 21 23 


27 


.4540 


.4566 


.4592 


.4617 


.4643 


.4669 


.4695 


62 


358 


10 13 16 


18 21 23 


28 


.4695 


.4720 


.4746 


.4772 


.4797 


.4823 


.4848 


61 


358 


10 13 16 


18 21 23 


29 


.4848 


.4874 


.4899 


.4924 


.4950 


.4975 


.5000 


60 


358 


10 13 15 


18 20 23 


30 


.5000 


.5025 


.5050 


.5075 


.5100 


.5125 


.5150 


69 


358 


10 13 15 


18 20 23 


31 


.5150 


.5175 


.5200 


.5225 


.5250 


.5275 


.5299 


68 


358 


10 13 15 


18 20 23 


33 


.5299 


.5324 


.5348 


.5373 


.5398 


.5422 


.5446 


67 


358 


10 13 15 


18 20 23 


32 


.5446 


.5471 


.5495 


.5519 


.5544 


.5568 


.5592 


66 


357 


10 12 14 


17 19 22 


34 


.5592 


.5616 


.5640 


.5664 


.5688 


.5712 


.5736 


66 


357 


10 12 14 


17 19 22 


36 


.5736 


.5760 


.5783 


.5807 


.5831 


.5854 


.5878 


64 


257 


10 12 14 


17 19 22 


36 


.5878 


.5901 


.5925 


.5948 


.5972 


.5995 


.6018 


63 


257 


9 12 14 


16 18 21 


37 


.6018 


.6041 


.6065 


.6088 


.6111 


.6134 


.6157 


62 


257 


9 12 14 


16 18 21 


38 


.6157 


.6180 


.6202 


.6225 


.6248 


.6271 


.6293 


61 


257 


9 12 14 


16 18 21 


39 


.6293 


.6316 


.6338 


.6361 


.6383 


.6406 


.6428 


60 


247 


9 11 13 


15 18 20 


10 


.6428 


.6450 


.6472 


.6494 


.6517 


.6539 


.6561 


49 


247 


9 11 13 


15 18 20 


LI 


.6561 


.6583 


.6604 


.6626 


.6648 


.6670 


.6691 


48 


247 


9 11 13 


15 18 20 


12 


.6691 


.6713 


.6734 


.6756 


.6777 


.6799 


.6820 


47 


246 


9 11 13 


15 17 19 


13 


.6820 


.6841 


.6862 


.6884 


.6905 


.6926 


.6947 


46 


246 


8 11 13 


15 17 19 


14 


.6947 


.6967 


.6988. 


.7009 


.7030 


.7050 


.7071 


46 


246 


8 11 13 


15 17 19 


O 




60' 


40' 


30' 


20' 


10' 


0' 





1' r 3' 


4' 6' 6' 


r ' v 



Natural Cosines 45-90 c 



Mean Differences 

(Subtract) 



TABLES OF USEFUL DATA 



449 



TABLE V. TRIGONOMETRIC FUNCTIONS. (Continued) 



Natural Sines 45-90 


Mean Differences 




(Add) 





0' 


10' 


20' 


30' 


40' 


60' 




o 


1' V V 


V 5' 6' 


V V & 


45 


.7071 


.7092 


.7112 


.7133 


.7153 


.7173 


.7193 


44 


246 


8 10 12 


14 16 18 


46 


.7193 


.7214 


.7234 


.7254 


.7274 


.7294 


.7314 


43 


246 


8 10 12 


14 16 18 


47 


.7314 


.7333 


.7353 


.7373 


.7392 


.7412 


.7431 


42 


246 


8 10 12 


14 15 18 


48 


.7431 


. 7451 


.7470 


.7490 


.7509 


.7528 


.7547 


41 


246 


8 10 11 


13 15 17 


49 


.7647 


.7566 


.7585 


.7604 


.7623 


.7642 


.7660 


40 


246 


8 9 11 


13 15 17 


60 


7660 


.7679 


. 7698 


.7716 


.7735 


.7753 


.7771 


39 


246 


7 9 11 


13 15 17 


61 


.7771 


.7790 


.7808 


.7826 


.7844 


.7862 


.7880 


38 


245 


7 9 11 


13 15 16 


62 


.7880 


.7898 


.7916 


.7934 


.7951 


.7969 


.7986 


37 


235 


7 9 10 


12 14 16 


63 


. 7986 


.8004 


.8021 


.8039 


.8056 


.8073 


.8090 


36 


235 


7 9 10 


12 14 16 


64 


.8090 


8107 


.8124 


.8141 


.8158 


.8175 


.8192 


36 


235 


7 8 10 


12 14 15 


66 


.8192 


. 8208 


.8225 


.8241 


.8258 


.8274 


.8290 


34 


235 


7 8 10 


11 13 15 


66 


.8290 


.8307 


.8323 


.8339 


.8355 


.8371 


.8387 


33 


235 


6 8 10 


11 13 15 


57 


.8387 


.8403 


.8418 


.8434 


.8450 


.8465 


.8480 


32 


235 


689 


11 12 14 


58 


.8480 


. 8496 


.8511 


.8526 


.8542 


.8557 


.8572 


31 


235 


689 


11 12 14 


59 


8572 


.8587 


.8601 


.8616 


.8631 


.8646 


.8660 


30 


134 


679 


10 12 13 


60 


.8660 


.8675 


.8689 


.8704 


.8718 


.8732 


.8746 


29 


134 


679 


10 11 13 


61 


.8746 


.8760 


.8774 


.8788 


.8802 


.8816 


.8829 


28 


134 


678 


10 11 12 


62 


.8829 


.8843 


.8857 


.8870 


.8884 


.8897 


.8910 


27 


1 3 4 


578 


9 11 12 


63 


.8910 


.8923 


.8936 


.8949 


.8962 


.8975 


.8988 


26 


134 


568 


9 10 12 


64 


.8988 


.9001 


.9013 


.9026 


.9038 


.9051 


.9063 


26 


134 


567 


9 10 11 


65 


.9063 


.9075 


.9088 


.9100 


.9112 


.9124 


.9135 


24 


124 


567 


9 10 11 


66 


.9135 


.9147 


.9159 


.9171 


.9182 


.9194 


.9205 


23 


124 


567 


8 9 11 


67 


.9205 


.9216 


.9228 


.9239 


.9250 


.9261 


.9272 


22 


123 


467 


8 9 10 


68 


.9272 


.9283 


. 9293 


9304 


.9315 


.9325 


.9336 


21 


123 


456 


7 9 10 


69 


.9336 


.9346 


.9356 


.9367 


.9377 


.9387 


.9397 


20 


123 


456 


789 


70 


9397 


.9407 


.9417 


.9426 


.9436 


.9446 


.9455 


19 


123 


456 


789 


71 


.9455 


.9465 


.9474 


.9483 


.9492 


.9502 


.9511 


18 


123 


456 


778 


72 


.9511 


. 9520 


.9528 


.9537 


9546 


.9555 


.9563 


17 


123 


345 


678 


73 


.9563 


.9572 


.9580 


.9588 


.9596 


.9605 


.9613 


16 


122 


345 


677 


74 


.9613 


.9621 


.9628 


.9636 


.9644 


.9652 


.9659 


15 


122 


345 


567 


76 


.9659 


.9667 


. 9674 


.9681 


.9689 


.9696 


.9703 


14 


112 


344 


567 


76 


.9703 


.9710 


.9717 


.9724 


.9730 


.9737 


.9744 


13 


1 1 2 


344 


566 


77 


.9744 


.9750 


.9757 


.9763 


.9769 


.9775 


.9781 


12 


112 


334 


456 


78 


.9781 


.9787 


.9793 


.9799 


.9805 


.9811 


.9816 


11 


112 


234 


455 


79 


.9816 


.9822 


.9827 


.9833 


.9838 


.9843 


.9848 


10 


1 1 2 


233 


445 


80 


.9848 


.9853 


.9858 


.9863 


.9868 


.9872 


.9877 


9 


1 1 


223 


344 


81 


.9877 


.9881 


.9886 


.9890 


.9894 


.9899 


.9903 


8 


1 1 


223 


344 


82 


.9903 


.9907 


.9911 


.9914 


.9918 


.9922 


.9925 


7 


1 1 


1 2 2 


333 


83 


.9925 


.9929 


.9932 


.9936 


.9939 


.9942 


.9945 


6 


1 1 


1 2 2 


233 


84 


.9945 


.9948 


.9951 


.9954 


.9957 


.9959 


.9962 


6 


1 1 


1 1 2 


223 


86 


. 9962 


. 9964 


. 9967 


.9969 


.9971 


.9974 


.9976 


4 


1 1 


1 1 2 


222 


86 


.9976 


.9978 


.9980 


.9981 


.9983 


.9985 


.9986 


3 


1 


1 1 1 


222 


87 


.9986 


.9988 


.9989 


.9990 


.9992 


.9993 


.9994 


2 


1 


1 1 1 


112 


88 


.9994 


.9995 


.9996 


.9997 


.9997 


.9998 


.9998 


1 


000 


1 


1 1 1 


89 


.9998 


.9999 


.9999 


1.0000 


1.0000 


1.0000 


1.0000 





000 


1 


111 


o 




60' 


40' 


30' 


20' 


10' 


0' 


o 


1' 2' 3' 


4' 5' 6' 


r v v 


Natural Cosines 0-45 


Mean Differences 

/Cj .Vi,.4--vn jt4-\ 




(Subtract) 



450 



LIGHT: PRINCIPLES. AND EXPERIMENTS 



TABLE V. TRIGONOMETRIC FUNCTIONS. (Continued) 



Natural Tangents 0-45 


Mean Differences 
(Add) 


O 


0' 


10' 


20' 


30' 


40' 


50' 




O 






V V V 









.0000 


.0029 


.0058 


.0087 


.0116 


.0145 


.0175 


89 


369 


12 15 17 


20 23 26 


1 


.0175 


.0204 


.0233 


.0262 


.0291 


.0320 


.0349 


88 


360 


12 15 17 


20 23 26 


2 


.0349 


.0378 


.0407 


.0437 


.0466 


.0495 


.0524 


87 


369 


12 15 17 


20 23 26 


3 


.0524 


.0553 


.0582 


.0612 


.0641 


.0670 


.0699 


86 


369 


12 15 17 


20 23 26 


4 


.0699 


.0729 


.0758 


.0787 


.0816 


.0846 


.0875 


86 


369 


12 15 18 


21 24 26 


5 


.0875 


.0904 


.0934 


.0963 


.0992 


.1022 


. 1051 


84 


369 


12 15 18 


21 24 26 


6 


.1051 


.1080 


.1110 


.1139 


.1169 


.1198 


.1228 


83 


369 


12 15 18 


21 24 27 


7 


.1228 


.1257 


.1287 


.1317 


.1346 


.1376 


.1405 


82 


369 


12 15 18 


21 24 27 


8 


.1405 


.1435 


.1465 


.1495 


.1524 


.1554 


.1584 


81 


369 


12 15 18 


21 24 27 


9 


.1584 


.1614 


.1644 


.1673 


.1703 


.1733 


.1763 


80 


369 


12 15 18 


21 24 27 


10 


.1763 


.1793 


.1823 


.1853 


.1883 


.1914 


.1944 


79 


369 


12 15 18 


21 24 27 


11 


.1944 


.1974 


.2004 


.2035 


.2065 


.2095 


.2126 


78 


369 


12 15 18 


21 24 27 


12 


.2126 


.2156 


.2186 


.2217 


.2247 


.2278 


.2309 


77 


369 


12 15 18 


22 25 28 


13 


.2309 


.2339 


.2370 


.2401 


.2432 


.2462 


.2493 


76 


369 


12 15 18 


22 25 28 


14 


.2493 


.2524 


.2555 


.2586 


.2617 


.2648 


.2679 


76 


369 


12 16 19 


22 25 28 


16 


.2679 


.2711 


.2742 


.2773 


.2805 


.2836 


.2867 


74 


369 


12 16 19 


22 25 28 


16 


.2867 


.2899 


.2931 


.2962 


.2994 


.3026 


.3057 


73 


3 6 10 


13 16 19 


22 26 29 


17 


.3057 


.3089 


.3121 


.3153 


.3185 


.3217 


.3249 


72 


3 6 10 


13 16 19 


22 26 29 


18 


.3249 


.3281 


.3314 


.3346 


.3378 


.3411 


.3443 


71 


3 6 10 


13 16 19 


23 26 29 


19 


.3443 


.3476 


.3508 


.3541 


.3574 


.3607 


.3640 


70 


3 7 10 


13 16 20 


23 26 30 


20 


.3640 


.3673 


.3706 


.3739 


.3772 


.3805 


.3839 


69 


3 7 10 


13 17 20 


23 26 30 


21 


.3839 


.3872 


.3906 


.3939 


.3973 


.4006 


.4040 


68 


3 7 10 


13 17 20 


23 27 30 


22 


.4040 


.4074 


.4108 


.4142 


.4176 


.4210 


.4245 


67 


3 7 10 


14 17 21 


24 27 31 


23 


.4245 


.4279 


.4314 


.4348 


.4383 


.4417 


.4452 


66 


3 7 10 


14 17 21 


24 28 31 


24 


.4452 


.4487 


.4522 


.4557 


.4592 


.4628 


.4663 


66 


4 7 11 


14 18 21 


25 28 32 


26 


.4663 


.4699 


.4734 


.4770 


.4806 


.4841 


.4877 


64 


4 7 11 


14 18 21 


25 29 32 


26 


.4877 


.4913 


.4950 


.4986 


.5022 


.5059 


.5095 


63 


4 7 11 


15 18 22 


25 29 33 


27 


.5095 


.5132 


.5169 


.5206 


.5243 


.5280 


.5317 


62 


4 7 11 


15 19 22 


26 30 33 


28 


.5317 


.5354 


.5392 


.5430 


.5467 


.5505 


.5543 


61 


4 8 11 


15 19 23126 30 34 


29 


.5543 


.5581 


.5619 


.5658 


.5696 


.5735 


.5774 


60 


4 8 12 


15 19 23 


27 31 35 


30 


.5774 


.5812 


.5851 


.5890 


.5930 


.5969 


.6009 


69 


4 8 12 


16 20 23 


27 31 35 


31 


.6009 


.6048 


.6088 


.6128 


.6168 


.6208 


.6249 


68 


4 8 12 


16 20 24 


28 32 36 


32 


.6249 


.6289 


.6330 


.6371 


.6412 


.6453 


.6494 


67 


4 8 12 


16 20 24 


28 33 37 


33 


.6494 


.6536 


.6577 


.6619 


.6661 


.6703 


.6745 


66 


4 8 13 


17 21 25 


29 34 38 


34 


.6745 


.6787 


.6830 


.6873 


.6916 


.6959 


.7002 


66 


4 9 13 


17 21 26 


30 34 39 


36 


.7002 


.7046 


.7089 


.7133 


.7177 


.7221 


.7265 


64 


4 9 13 


18 22 26 


31 35 40 


36 


.7265 


.7310 


.7355 


.7400 


.7445 


.7490 


.7536 


63 


5 9 14 


18 23 27 


32 36 41 


37 


.7536 


.7581 


.7627 


.7673 


.7720 


.7766 


.7813 


62 


5 9 14 


18 23 28 


32 37 42 


38 


.7813 


.7860 


.7907 


.7954 


.8002 


.8050 


.8098 


61 


5 10 14 


19 24 29 


33 38 43 


39 


.8098 


.8146 


.8195 


.8243 


.8292 


.8342 


.8391 


60 


5 10 15 


20 24 29 


34 39 44 


40 


.8391 


.8441 


.8491 


.8541 


.8591 


.8642 


.8693 


49 


5 10 15 


20 25 30 


35 40 45 


41 


.8693 


.8744 


.8796 


.8847 


.8899 


.8952 


.9004 


48 


5 10 16 


21 26 31 


36 42 47 


42 


.9004 


.9057 


.9110 


.9163 


.9217 


.9271 


.9325 


47 


6 11 16 


21 27 32 


37 43 48 


43 


.9325 


.9380 


.9435 


.9490 


.9545 


.9601 


.9657 


46 


6 11 17 


22 28 33 


39 44 50 


44 


.9657 


.9713 


.9770 


.9827 


.9884 


.9942 


1.0000 


46 


6 11 17 


23 29 34 


40 46 51 


o 




50' 


40' 


30' 


20' 


10' 


0' 





1' V 3' 


4' 5' 6' 


r v v 


Natural Cotangents 45-90 


Mean Differences 

(Subtract) 



TABLES OF USEFUL DATA 



451 



TABLE V. - -TRIGONOMETRIC FUNCTIONS. (Continued) 



Natural Tangents 45-90 



Differences 

















0' 10' 20' 30' 40' 50' 


o 


0' 


10' 


20' 30' 


40' 


50' 







to to 


to to to to 




















10' 20' 


30 40' 


50' 60' 


45 


1.0000 


1 .0058 


1.0117 


1.0176 


1.0235 


1.0295 


1.0355 


44 


58 59 


59 59 


60 60 


46 


.0355 


.0416 


.0477 


.0538 


.0599 


.0661 


.0723 


43 


61 61 


61 61 


62 62 


47 


.0723 


.0786 


.0850 


.0913 


.0977 


.1041 


.1106 


42 


63 64 


63 64 


64 65 


48 


.1106 


.1171 


.1237 


.1303 


.1369 


.1436 


.1504 


41 


65 66 


66 66 


67 68 


49 


.1504 


.1572 


.1640 


.1709 


.1778 


.1847 


.1918 


40 


68 68 


69 69 


69 71 


50 


.1918 


.1988 


.2059 


.2131 


.2203 


.2276 


.2349 


39 


70 71 


72 72 


73 73 


61 


.2349 


.2423 


.2497 


.2572 


.2647 


.2723 


.2799 


38 


74 74 


75 75 


76 76 


52 


.2799 


.2876 


.2954 


.3032 


.3111 


.3190 


.3270 


37 


77 78 


78 79 


79 80 


53 


.3270 


.3351 


.3432 


.3514 


.3597 


.3680 


.3764 


36 


81 81 


82 83 


83 84 


54 


.3764 


.3848 


.3933 


.4019 


.4106 


.4193 


.4281 


36 


84 85 


86 87 


87 88 


55 


.4281 


.4370 


.4460 


.4550 


.4641 


.4733 


.4826 


34 


89 90 


90 91 


92 93 


56 


.4826 


.4919 


.5013 


.5108 


.5204 


.5301 


.5399 


33 


93 94 


95 96 


97 98 


67 


.5399 


.5497 


.5597 


.5697 


.5798 


.5900 


.6003 


32 


98 100 


100 101 


102 103 


68 


.6003 


.6107 


.6212 


.6319 


.6426 


.6534 


.6643 


31 


104 105 


107 107 


108 109 


59 


.6643 


.6753 


.6864 


.6977 


.7090 


.7205 


.7321 


30 


110 111 


113 113 


115 116 


60 


.7321 


.7437 


.7556 


.7675 


.7796 


.7917 


.8040 


29 


116 119 


119 121 


121 123 


61 


.8040 


.8165 


.8291 


.8418 


.8546 


.8676 


.8807 


28 


125 126 


127 128 


130 131 


62 


.8807 


.8940 


.9074 


1.9210 


1.9347 


1.9486 


1.9626 


27 


133 134 


136 137 


139 140 


63 


1.9626 


1.9768 


1.9912 


2.0057 


2.0204 


2.0353 


2.0503 


26 


142 144 


145 147 


149 150 


64 


2.0503 


2.0655 


2.0809 


.0965 


.1123 


.1283 


.1445 


26 


152 154 


156 158 


160 162 


66 


.1445 


.1609 


.1775 


.1943 


.2113 


.2286 


.2460 


24 


164 166 


168 17oli73 174 


66 


.2460 


.2637 


.2817 


.2998 


.3183 


.3369 


.3559 


23 


177 180 


181 185 


186 190 


67 


.3559 


.3750 


.3945 


.4142 


.4342 


.4545 


.4751 


22 


191 195 


197 200 


203 206 


68 


.4751 


.4960 


.5172 


.5386 


.5605 


.5826 


.6051 


21 


209 212 


214 219 


221 225 


69 


.6051 


.6279 


.6511 


.6746 


.6985 


.7228 


.7475 


20 


228 232 


235 239 


243 247 


70 


.7475 


.7725 


.7980 


.8239 


2.8502 


2.8770 


2.9042 


19 


250 255 


259 263 


268 272 


71 


2.9042 


2.9319 


2.9600 


2.9887 


3.0178 


3.0475 


3.0777 


18 


277 281 


287 291 


297 302 


72 


3.0777 


3.1084 


3.1397 


3.1716 


.2041 


.2371 


.2709 


17 


307 313 


319 325 


330 338 


73 


.2709 


.3052 


.3402 


.3759 


.4124 


.4495 


.4874 


16 


343 350 


357 365 


371 379 


74 


.4874 


.5261 


.5656 


.6059 


.6470 


.6891 


3.7321 


15 


387 395 


403 411 


421 430 


75 


3.7321 


3.7760 


3.8208 


3.8667 


3.9136 


3.9617 


4.0108 


14 


439 448 


459 469 


481 491 


76 


4.0108 


4.0611 


4.1126 


4.1653 


4.2193 


4.2747 


.3315 


13 


503 515 


527 540 


554 568 


77 


.3315 


.3897 


.4494 


.5107 


.5736 


4.6382 


4.7046 


12 


582 597 


613 629 


646 664 


78 


4.7046 


4.7729 


4.8430 


4.9152 


4.9894 


5.0658 


5.1446 


11 


683 701 


722 742 


764 788 


79 


5.1446 


5.2257 


5.3093 


5.3955 


5.4845 


5.5764 


5.6713 


10 


811 836 


862 890 


919 949 


80 


5.6713 


5.7694 


5.8708 


5.9758 


6.0844 


6.1970 


6.3138 


9 




81 


6.3138 


6.4348 


6.5606 


6.6912 


6.8269 


6.9682 


7.1154 


8 




82 


7.1154 


7.2687 


7.4287 


7.5958 


7.7704 


7.9530 


8.1443 


7 




83 


8.1443 


8.3450 


8.5555 


8.7769 


9.0098 


9.2553 


9.5144 


6 




84 


9.5144 


9.7882 


10.078 


10.385 


10.712 


11.059 


11.430 


6 


Differences not 


85 


11.430 


11.826 


12.251 


12.706 


13.197 


13.727 


14.301 


4 


sufficiently 


86 


14.301 


14.924 


15.605 


16.350 


17.169 


18.075 


19.081 


3 


accurate 


87 


19.081 


20.206 


21.470 


22.904 


24.542 


26.432 


28.636 


2 




88 


28.636 


31.242 


34.368 


38.188 


42.964 


49.104 


57.290 


1 




89 


57.290 


68.750 


85.940 


114.59 


171.89 


343.77 


00 












60' 


40' 


30' 


20' 


10' 


0' 















1 








Natural Cotangents 0-45 





452 



LIGHT: PRINCIPLES AND EXPKRIMKNTX 



TABLE VI. LOGARITHMS OK TRIGONOMETRY FUNCTIONS 



Logarithmic Sines 0-45 


Mean Differences 
(Add) 


Q 


0' 


10' 


20' 


30' 


40' 


50' 




O 


1' 2' 3' 


4' 5' 


6' 


r 8' 9' 





oo 


3.4637 


.7648 


.9408 .0658 


.1627 


.2419 


89 




3011-792 


1 


2.2419 


.3088 


.3668 


.4179 


.4637 


.5050 


.5428 


88 




669-378 


2 


.5428 


.5776 


.6097 


.6397 


.6677 


.6940 


.7188 


87 




348-248 


3 


.7188 


.7423 


.7645 


.7857 


.8059 


.8251 


.8436 


86 




235-185 


4 


.8436 


.8613 


.8783 


.8946 


.9104 


.9256 


.9403 


86 


Differences 


177-147 


5 


2.9403 


.9545 


.9682 


.9816 


.9945 


.0070 


.0192 


84 




142-122 


6 


1.0192 


.0311 


.0426 


.0539 


.0648 


.0755 


.0859 


83 




119-104 


7 


.0859 


.0961 


.1060 


.1157 


. 1252 


. 1345 


.1436 


82 




102-91 


8 


.1436 


.1525 


.1612 


.1697 


.1781 


.1863 


.1943 


81 




89-80 


9 


.1943 


.2022 


.2100 


.2176 


.2251 


.2324 


.2397 


80 




79-73 


10 


1 . 2397 


.2468 


. 2538 


.2606 


.2674 


.2740 


.2806 


79 


7 14 20 27 34 


41 


48 54 61 


11 


.2806 


.2870 


.2934 


.2997 


.3058 


.3119 


.3179 


78 


6 12 19 


25 31 


37 


43 50 56 


12 


.3179 


.3238 


.3296 


.3353 


.3410 


. 3466 


.3521 


77 


6 11 17 


23 29 


34 


40 46 51 


13 


.3521 


.3575 


3629 


.3682 


.3734 


.3786 


.3837 


76 


5 11 16 


21 27 


32 


37 42 48 


14 


.3837 


.3887 


3937 


.3986 


.4035 


.4083 


.4130 


75 


5 10 15 


20 25 


29 


34 39 44 


15 


T.4130 


.4177 


.4223 


.4269 


.4314 


.4359 


.4403 


74 


5 9 14 


18 23 


27 


32 37 41 


16 


.4403 


.4447 


.4491 


.4533 


.4576 


4618 


.4659 


73 


4 9 13 


17 21 


26 


30 34 39 


17 


.4659 .4700 


.4741 


.4781 


.4821 


.4861 


.4900 


72 


4 8 12 


16 20 


24 


28 32 36 


18 .4900 


.4939 


.4977 


.5015 


.5052 


. 5090 


5126 


71 


4 8 11 


15 19 


23 


27 30 34 


19 .5126 


.5163 


5199 


. 5235 


.5270 


. 5306 


.5341 70 


4 7 10 


14 18 


22 


25 29 32 


20 T.5341 


.5375 


5409 


.5443 


. 5477 


.5510 


.5543 69 


3 7 10 


14 17 


2024 27 31 


21 


.5543 


.5576 


5609 


.5641 


5673 


5704 


.5736 68 


3 6 10 


13 16 


19 22 26 29 


22 


. 5736 


.5767 


5798 


.5828 


.5859 


.5889 


.5919 


67 


369 


12 15 


18 


21 24 28 


23 


.5919 


.5948 


. 5978 


6007 


.6036 


6065 


.6093 


66 


369 


12 15 


17 


20 23 26 


24 


.6093 


.6121 


6149 


.6177 


.6205 


.6232 


.6259 


65 


368 


11 14 


16 


19 22 25 


25 


T.6259 


.6286 


6313 


6340 


.6366 


.6392 


6418 


64 


358 


11 13 


16 


19 21 24 


26 


.6418 


.6444 


.6470 


.6495 


.6521 


.6546 


.6570 


63 


358 


10 13 


15 


18 20 23 


27 6570 


6595 


6620 


.6644 


.6668 


.6692 


.6716 


62 


257 


10 12 


15 


17 19 22 


28 .6716 


.6740 


6763 


.6787 


.6810 


.6833 


6856 


61 


257 


9 12 


14 16 19 21 


29 


.6856 


.6878 


.6901 


.6923 


.6946 


.6968 


.6990 


60 


247 


9 11 


13 


16 18 20 


30 


I. 6990 


.7012 


.7033 


.7055 


.7076 


.7097 


.7118 


69 


246 


9 11 


13 


15 17 19 


31 


.7118 


.7139 


.7160 


.7181 


.7201 


7222 


.7242 


68 


246 


8 10 


12 


14 16 19 


32 


.7242 


.7262 


.7282 


.7302 


.7322 


.7342 


.7361 


67 


246 


8 10 


12 


14 16 18 


33 


.7361 


.7380 


.7400 


.7419 


.7438 


.7457 


.7476 


66 


246 


8 10 


11 


13 15 17 


34 


.7476 


.7494 


.7513 


.7531 


.7550 


.7568 


. 7586 


65 


246 


7 9 


11 


13 15 16 


36 


I. 7586 


.7604 


.7622 


.7640 


.7657 


.7675 


.7692 


54 


245 


7 9 


11 


12 14 16 


36 


.7692 


.7710 


.7727 


.7744 


.7761 


.7778 


.7795 


53 


235 


7 9 


10 


12 14 15 


37 


7795 


.7811 


.7828 


.7844 


.7861 


.7877 


.7893 


52 


235 


7 8 


10 


11 13 15 


38 


.7893 


.7910 


.7926 


.7941 


.7957 


.7973 


.7989 


61 


235 


6 8 


10 


11 13 14 


39 


.7989 


.8004 


.8020 


.8035 


.8050 


.8066 


.8081 


60 


235 


6 8 


9 


11 12 14 


40 


T.8081 


.8096 


.8111 


.8125 


.8140 


.8155 


.8169 


49 


1 3 4 


6 7 


9 


10 12 13 


41 


.8169 


.8184 


.8198 


.8213 


.8227 


.8241 


.8255 


48 


1 3 4 


6 7 


9 


10 11 13 


42 


.8255 


.8269 


.8283 


.8297 


.8311 


.8324 


.8338 


47 


1 3 4 


6 7 


8 


10 11 13 


43 


.8338 


.8351 


.8365 


.8378 


.8391 


.8405 


.8418 


46 


1 3 4 


5 7 


8 


9 11 12 


44 


.8418 


.8431 


.8444 


.8457 


.8469 


.8482 


.8495 


46 


1 3 4 


5 6 


8 


9 10 12 


O 




60' 


40' 


30' 


20' 


10' 


0' 





1' 2' 3' 


4' 6' 


6' 


r v 9' 


Logarithmic Cosines 45-90 


Mean Differences 
(Subtract) 



TABLES OF USEFUL DATA 



453 



TABLE VI. LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. (Continued) 



Logarithmic Sines 45-90 


Mean Differences 




(Subtract) 


O 


0' 


10' 


20' 


30' 


40' 


60' 




O 


i' 2' 3' 


4 5 6' 


T 8' 9 


45 


T.8495 


.8507 


.8520 


.8532 


.8545 


.8557 


.8569 


44 


1 2 4 


567 


9 10 11 


46 


.8569 


.8582 


.8594 


.8606 


.8618 


.8629 


.8641 


43 


1 2 4 


567 


8 10 11 


47 


.8641 


.8653 


.8665 


.8676 


.8688 


.8699 


.8711 


42 


1 2 4 


5 6 7 


8 9 11 


48 


.8711 


.8722 


.8733 


.8745 


.8756 


.8767 


.8778 


41 


1 2 3 


467 


8 9 1C 


49 


.8778 


.8789 


.8800 


.8810 


.8821 


.8832 


.8843 


40 


I 2 3 


457 


8 1C 


60 


1.8843 


.8853 


.8864 


.8874 


.8884 


.8895 


.8905 


39 


1 2 3 


450 


7 9 1C 


61 


.8905 


.8915 


.8925 


.8935 


.8945 


.8955 


.8965 


38 


1 2 3 


456 


7 8 1 


62 


.8965 


.8975 


.8985 


.8995 


.9004 


.9014 


.9023 


37 


1 2 3 


456 


7 8 I 


63 


. 9023 


.9033 


.9042 


.9052 


.9061 


.9070 


.9080 


36 


1 2 3 


456 


7 8 < 


64 


.9080 


.9089 


.9098 


.9107 


.9116 


.9125 


.9134 


36 


1 2 3 


445 


6 7 I 


66 


1.9134 


.9142 


.9151 


.9160 


.9169 


.9177 


.9186 


34 


1 2 3 


345 


6 7 * 


66 


.9186 


.9194 


.9203 


.9211 


.9219 


.9228 


.9236 


33 


122 


345 


6 7 7 


67 


.9236 


.9244 


.9252 


.9260 


.9268 


.9276 


.9284 


32 


1 2 2 


345 


667 


68 


.9284 


.9292 


.9300 


.9308 


.9315 


.9323 


.9331 


31 


1 2 2 


345 


6 6 7 


69 


.9331 


.9338 


.9346 


.9353 


.9361 


.9368 


.9375 


30 


122345 


5 6 7 


60 


1 . 9375 


.9383 


.9390 


.9397 


.9404 


.9411 


.9418 


29 


112344 


5 6 f 


61 


.9418 


9425 


9432 


.9439 


.9446 


.9453 


.9459 


28 


1 1 2! 3 3 4 


5 5 < 


62 


.9459 


.9466 


.9473 


.9479 


.9486 


.9492 


.9499 


27 


1 2334 


5 5 f 


63 


.9499 


.9505 


9512 


.9518 


.9524 


9530 


.9537 


26 


1 2334 


4 5 < 


64 


.9537 


.9543 


.9549 


.9555 


.9561 


9567 


.9573 


26 


1 2234 


4 5 


66 


1.9573 


.9579 


.9584 


9590 


.9596 


.9602 


.9607 


24 


1 2233 


4 5 


66 


.9607 


9613 


9618 


9624 


.9629 


.9635 


.9640 


23 


1 2 


233 


4 4 t 


67 


.9640 


9646 


.9651 


.9656 


.9661 


.9667 


.9672 


22 


1 2 


2 3 3 


4 4 i 


68 


.9672 


.9677 


.9682 


.9687 


.9692 


9697 


.9702 


21 


1 2 


233 


4 4 


69 


.9702 


.9706 


.9711 


.9716 


.9721 


.9725 


.9730 


20 


1 


223 


3 4 4 


70 


T.9730 


.9734 


.9739 


.9743 


.9748 


.9752 


.9757 


19 


1 


223 


4 4 A 


71 


.9757 


.9761 


.9765 


.9770 


.9774 


.9778 


.9782 


18 


1 


222 


3 3 A 


72 


.9782 


.9786 


.9790 


.9794 


.9798 


.9802 


.9806 


17 


V 2 2 2 


3 3 4 


73 


.9806 


.9810 


.9814 


.9817 


.9821 


.9825 


.9828 


16 


1 


1 2 2 


3 3 ; 


74 


.9828 


.9832 


.9836 


.9839 


.9843 


.9846 


.9849 


16 


1 


1 2 2 


2 3 2 


76 


1.9849 


9853 


.9856 


.9859 


.9863 


.9866 


.9869 


14 


1 1 


122 


2 3 ? 


76 


.9869 


.9872 


.9875 


.9878 


.9881 


.9884 


.9887 


13 


1 1 


1 2 2 


2 2 ^ 


77 


.9887 


.9890 


.9893 


.9896 


.9899 


.9901 


.9904 


12 


1 1 


1 1 2 


2 2 ? 


78 


.9904 


.9907 


.9909 


.9912 


.9914 


.9917 


.9919 


11 


I 1 


1 1 2 


2 2 ^ 


79 


.9919 


.9922 


.9924 


.9927 


.9929 


.9931 


.9934 


10 


I 1 


1 1 2 


2 2 2 


80 


1.9934 


.9936 


.9938 


.9940 


.9942 


.9944 


.9946 


9 


2 


81 


.9946 


.9948 


.9950 


.9952 


.9954 


.9956 


.9958 


8 


2 


82 


.9958 


.9959 


.9961 


.9963 


.9964 


.9966 


.9968 


7 


1-2 


83 


.9968 


.9969 


.9971 


.9972 


.9973 


.9975 


.9976 


6 


1-2 


84 


.9976 


.9977 


.9979 


.9980 


.9981 


.9982 


.9983 


6 


1-2 


86 


1.9983 


.9985 


.9986 


.9987 


.9988 


.9989 


.9989 


4 


Differences t) 


86 


.9989 


.9990 


.9991 


.9992 


.9993 


.9993 


.9994 


3 


0-1 


87 


.9994 


.9995 


.9995 


.9996 


.9996 


.9997 


.9997 


2 


0-1 


88 


.9997 


.9998 


.9998 


.9999 


.9999 


.9999 


.9999 


1 


01 


89 


.9999 


.0000 


.0000 


.0000 


.0000 


.0000 


.0000 





O-l 


o 




60' 


40' 


30' 


20' 


10' 


0' 


O 


1' V 3' 


4 5' V 


7' 8' 9' 


Logarithmic Cosines 0-45 


Mean Differences 
(Subtract) 



454 



LIGHT: PRINCIPLES AND EXPERIMENTS 



TABLE VI. LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. (Continued) 



Logarithmic Tangents 0-45 


Mean Differences 
(Add) 


o 


0' 


10' 


20' 


30' 


40' 


50' 



















6' 







oo 


3.4637 


.7648 


.9409 


,0668 


. 1687 


.8419 


89 




3011-792 


1 


2.2419 


.3089 


.3669 


.4181 


.4638 


.5053 


.5431 


88 




670 378 


2 


.5431 


.5779 


.6101 


.6401 


.6682 


.6945 


.7194 


87 




348-249 


3 


.7194 


.7429 


.7652 


.7865 


.8067 


.8261 


.8446 


86 




235-185 


4 


.8446 


.8624 


.8795 


.8960 


.9118 


9272 


.9420 


86 




170-148 




















Differences 




5 


2.9420 


.9563 


.9701 


.9836 


.9966 


.0093 


.0816 


84 




143-124 


6 


1.0216 


.0336 


.0453 


.0567 


.0678 


.0786 


.0891 


83 




120-105 


7 


.0891 


.0995 


.1096 


.1194 


.1291 


.1385 


.1478 


82 




104-93 


8 


.1478 


.1569 


.1658 


.1745 


.1831 


.1915 


.1997 


81 




91-82 


9 


.1997 


.2078 


.2158 


.2236 


.2313 


.2389 


.2463 


80 




81-74 


10 


T.2463 


.2536 


.2609 


.2680 


.2750 


.2819 


.2887 


79 


7 14 21 


28 36 


43 


50 57 64 


11 


.2887 


.2954 


.3020 


.3085 


.3149 


.3212 


.3275 


78 


7 13 20 


26 33 


39 


46 52 59 


12 


.3275 


.3336 


3397 


.3458 


.3517 


.3576 


.3634 


77 


6 12 18 


24 30 


36 


42 48 54 


13 


.3634 


.3691 


.3748 


.3804 


3859 


.3914 


.3968 


76 


6 11 17 


22 28 


33 


39 44 50 


14 


.3968 


.4021 


.4074 


.4127 


.4178 


.4230 


.4281 


76 


5 10 16 


21 26 


31 


36 42 47 


15 


1.4281 


.4331 


.4381 


.4430 


4479 


.4527 


.4575 


74 


5 10 15 


20 25 


29 


34 39 44 


16 


.4575 


.4622 


.4669 


.4716 


.4762 


.4808 


.4853 


73 


5 <J 14 


19 24 


28 


33 38 42 


17 


.4853 


.4898 


.4943 


.4987 


. 5031 


.5075 


.5118 


72 


4 9 13 


18 22 


27 


31 35 40 


18 


.5118 


.5161 


.5203 


. 5245 


.5287 


. 5329 


.5370 


71 


4 8 13 


17 21 


25 


29 34 38 


19 


.5370 


.5411 


.5451 


.5491 


. 5531 


.5571 


5611 


70 


4 8 12 


16 20 


24 


28 32 36 


20 


1.5611 


.5650 


.5689 


.5727 


.5766 


.5804 


.5842 


69 


4 8 12 


15 19 


23 


27 31 35 


21 


.5842 


.5879 


.5917 


5954 


.5991 


.6028 


.6064 


68 


4 7 11 


15 18 


22 


26 30 33 


22 


.6064 


.6100 


.6136 


.6172 


.6208 


.6243 


.6279 


67 


4 7 11 


14 18 


22 


25 29 32 


23 


.6279 


.6314 


.6348 


.6383 


.6417 


6452 


.6486 


66 


3 7 10 


14 17 


21 


24 28 31 


24 


.6486 


.6520 


.6553 


.6587 


.6620 


.6654 


.6687 


66 


3 7 10 


13 17 


20 


23 27 30 


26 


1.6687 


.6720 


.6752 


6785 


.6817 


.6850 


.6882 


64 


3 7 10 


13 16 


20 


23 26 29 


26 


6882 


.6914 


.6946 


6977 


.7009 


.7040 


.7072 


63 


3 6 10 


13 16 


19 


22 26 29 


27 


.7072 


.7103 


.7134 


.7165 


.7196 


.7226 


7257 


62 


3 6 9 


12 15 


18 


21 25 28 


28 


.7257 


.7287 


.7317 


.7348 


.7378 


.7408 


.7438 


61 


369 


12 15 


18 


21 24 27 


29 


.7438 


.7467 


.7497 


7526 


.7556 


.7585 


.7614 


60 


369 


12 15 


18 


21 23 20 


30 


1.7614 


.7644 


.7673 


7701 


.7730 


.7759 


.7788 


69 


369 


12 14 


17 


20 23 26 


31 


.7788 


.7816 


. 7845 


.7873 


.7902 


.7930 


.7958 


58 


369 


11 14 


17 


20 23 25 


32 


.7958 


.7986 


.8014 


8042 


.8070 


.8097 


.8125 


67 


368 


11 14 


17 


20 22 25 


33 


.8125 


.8153 


.8180 


.8208 


.8235 


.8263 


.8290 


56 


368 


11 14 


17 


19 22 25 


34 


.8290 


.8317 


.8344 


.8371 


.8398 


.8425 


.8452 


55 


358 


11 13 


16 


19 22 24 


35 


1.8452 


.8479 


.8506 


.8533 


.8559 


.8586 


.8613 


64 


3 5 8 


11 13 


16 


19 21 24 


36 


.8613 


.8639 


.8666 


.8692 


.8718 


.8745 


.8771 


63 


358 


11 13 


16 


18 21 24 


37 


.8771 


.8797 


.8824 


.8850 


.8876 


.8902 


.8928 


62 


358 


10 13 


16 


18 21 24 


38 


.8928 


.8954 


.8980 


.9006 


.9032 


.9058 


.9084 


61 


358 


10 13 


16 


18 21 23 


39 


.9084 


.9110 


.9135 


.9161 


.9187 


.9212 


.9238 


60 


358 


10 13 


15 


18 21 23 


40 


1.9238 


.9264 


.9289 


.9315 


.9341 


.9366 


.9392 


49 


358 


10 13 


15 


18 21 23 


41 


.9392 


.9417 


.9443 


.9468 


.9494 


.9519 


.9544 


48 


358 


10 13 


15 


18 20 23 


42 


.9544 


.9570 


.9595 


.9621 


.9646 


.9671 


.9697 


47 


358 


10 13 


15 


18 20 23 


43 


.9697 


.9722 


.9747 


.9772 


.9798 


.9823 


.9848 


46 


358 


10 13 


15 


18 20 23 


44 


.9848 


.9874 


.9899 


.9924 


.9949 


.9975 


.0000 


46 


358 


10 13 


15 


18 20 23 







60' 


40' 


30' 


20' 


10' 


0' 





1' V 3' 


4 5' 6' 


7' 8' 9' 


logarithmic Cotangents 45-90 


Mean Differences 
(Subtract) 



TABLES OF USEFUL DATA 



455 



TABLE VI. LOGARITHMS OF TRIGONOMETRIC FUNCTIONS. (Continued) 



logarithmic Tangents 45-90" 


^TJ.V,CHA x-'iAic;! ^riiv.fo 

(Add) 


O 


0' 


10' 


20' 


30' 


40' 


50' 







1' V V 


V V V 


7' 8' 9' 


45 


0.0000 


.0025 


.0051 


.0076 


.0101 


.0126 


.0152 


44 


358 


10 13 


15 


18 20 23 


46 


.0152 


.0177 


.0202 


.0228 


.0253 


.0278 


.0303 


43 


358 


10 13 


15 


18 20 23 


47 


.0303 


.0329 


.0354 


.0379 


.0405 


.0430 


.0456 


42 


358 


10 13 


15 


18 20 23 


48 


.0456 


.0481 


.0506 


.0532 


.0557 


.0583 


.0608 


41 


358 


10 13 


15 


18 20 23 


49 


.0608 


.0634 


.0659 


.0685 


.0711 


.0736 


.0762 


40 


358 


10 13 


15 


18 20 23 


50 


0.0762 


.0788 


.0813 


.0839 


.0865 


.0890 


.0916 


39 


358 


10 13 


15 


18 20 23 


51 


.0916 


.0942 


.0968 


.0994 


.1020 


.1046 


.1072 


38 


358 


10 13 


16 


18 21 23 


52 


.1072 


.1098 


.1124 


.1150 


.1176 


.1203 


.1229 


37 


358 


10 13 


16 


18 21 23 


53 


.1229 


.1255 


.1282 


.1308 


.1334 


.1361 


.1387 


36 


358 


11 13 


16 


18 21 24 


54 


.1387 


.1414 


.1441 


.1467 


.1494 


. 1521 


.1548 


35 


358 


11 13 


16 


19 21 24 


55 


1548 


.1575 


.1602 


.1629 


. 1656 


.1683 


.1710 


34 


358 


11 14 


16 


19 22 24 


56 


.1710 


.1737 


.1765 


.1792 


.1820 


.1847 


.1875 


33 


368 


11 14 


17 


19 22 25 


57 


.1875 


.1903 


.1930 


.1958 


.1986 


.2014 


.2042 


32 


368 


11 14 


17 


20 22 25 


58 


.2042 


.2070 


.2098 


.2127 


.2155 


.2184 


.2212 


31 


369 


11 14 


17 


20 23 35 


69 


.2212 


.2241 


.2270 


.2299 


.2327 


.2356 


.2386 


30 


369 


12 15 17 


20 23 26 


60 


2386 


.2415 


.2444 


.2474 


.2503 


2533 


.2562 


29 


369 


12 15 


18 


21 23 26 


61 


.2562 


.2592 


.2622 


.2652 


.2683 


.2713 


.2743 


28 


369 


12 15 


18 


21 24 27 


62 


.2743 


.2774 


.2804 


.2835 


.2866 


.2897 


.2928 


27 


369 


12 15 


19 


22 25 28 


63 


.2928 


.2960 


.2991 


.3023 


.3054 


.3086 


.3118 


26 


3 6 10 


13 16 


19 


22 25 29 


64 


.3118 


.3150 


.3183 


.3215 


.3248 


.3280 


.3313 


26 


3 7 10 


13 16 


20 


23 26 29 


65 


0.3313 


.3346 


.3380 


.3413 


.3447 


.3480 


.3514 


24 


3 7 10 


13 17 


20 


23 27 30 


66 


.3514 


.3548 


.3583 


.3617 


.3652 


.3686 


.3721 


23 


3 7 10 


14 17 


21 


24 28 31 


67 


.3721 


.3757 


.3792 


.3828 


.3864 


.3900 


.3936 


22 


4 7 11 


14 18 22 


25 29 32 


68 


.3936 


.3972 


.4009 


.4046 


.4083 


.4121 


.4158 


21 


4 7 11 


15 19 


22 


26 30 33 


69 


.4158 


.4196 


.4234 


.4273 


.4311 


.4350 


.4389 


20 


4 8 12 


15 19 


23 


27 31 35 


70 


0.4389 


.4429 


.4469 


.4509 


.4549 


.4589 


.4630 


19 


4 8 12 


16 20 24 


28 32 36 


71 


.4630 


.4671 


4713 


.4755 


.4797 


.4839 


.4882 


18 


4 8 13 


17 21 


25 


29 34 38 


72 


.4882 


.4925 


.4969 


.5013 


.5057 


.5102 


.5147 


17 


4 9 13 


18 22 


26 


31 35 40 


73 


.5147 


.5192 


.5238 


.5284 


.5331 


.5378 


.5425 


16 


5 9 14 


19 23 


28 


32 37 42 


74 


.5425 


.5473 


.5521 


.5570 


.5619 


.5669 


.5719 


16 


5 10 15 


20 25 


29 


34 39 44 


76 


0.5719 


.5770 


.5822 


.5873 


.5926 


.5979 


.6032 


14 


5 10 16 


21 26 


31 


37 42 47 


76 


.6032 


.6086 


.6141 


.6196 


.6252 


.6309 


.6366 


13 


6 11 17 


22 28 


33 


39 45 50 


77 


.6366 


.6424 


.6483 


.6542 


.6603 


.6664 


.6725 


12 


6 12 18 


24 30 


36 


42 48 54 


78 


.6725 


.6788 


.6851 


.6915 


.6981 


.7047 


.7113 


11 


7 13 20 


26 33 


39 


46 52 59 


79 


.7113 


.7181 


.7250 


.7320 


.7391 


.7464 


.7537 


10 


7 14 21 


28 35 


42 


49 57 64 


80 


0.7537 


.7611 


.7687 


.7764 


.7842 


.7922 


.8003 


9 




74-81 


81 


.8003 


.8085 


.8169 


.8255 


.8342 


.8431 


.8522 


8 




82-91 


82 


.8522 


.8615 


.8709 


.8806 


.8904 


.9005 


.9109 


7 




93-104 


83 


.9109 


.9214 


.9322 


.9433 


.9547 


.9664 


.9784 


6 




105-120 


84 


.9784 


.9907 


.0034 


.0164 


.0299 


.0437 


.0580 


6 


Differences 


123-143 


86 


1.0580 


.0728 


.0882 


.1040 


. 1205 


.1376 


.1554 


4 




148-178 


86 


.1554 


.1739 


.1933 


.2135 


.2348 


.2571 


.2806 


3 




185-235 


87 


.2806 


.3055 


.3318 


.3599 


.3899 


.4221 


.4569 


2 




249-348 


88 


.4569 


.4947 


.5362 


.5819 


.6331 


.6911 


.7581 


1 




378-670 


89 


.7581 


.8373 


.9342 


2.0591 


.2352 


.5363 


00 







792-301 1 


o 




50' 


40' 


30' 


20' 


10' 


0' 


o 


1' 2' 3' 


4 5' 6 


T 8' 9' 


Logarithmic Cotangents 0-45 


Mean Differences 
(Subtract) 



ANSWERS TO PROBLEMS 

Answers are included only for those problems requiring a numerical solution 

Chapter II, page 17 

1. x 9 = -6.67; y' = -5.0. 2. x' = -8; y' = +1. 

3. For a real image, 62.5 cm. away; for a virtual image, 37.5 cm. away. 

4. 72.25 cm. toward the mirror. 

5. 5. 6. 60 cleg. 

7. 1.5 cm. from surface; diameter = 1.20 mm.; a = 1.04. 

8. n = 2. 

9. 13.9 from the side; = +1.235; a = +1.15. 
10. About 54 ft. 

12. 78.9 from the emergent nodal point. 

Chapter III, page 29 

2. / = /' = +15; p = p' = 10; P and P' coincide at middle of sphere. 

3. / = /' = +20; p = 0; p' = -6.67. 

4. / = +15.9; /' - +21.2; A - -136; p = -<> 17 . 

6. ft = +23.266; // = +31.095; / 2 = + 132.85; // = +142.85; 
/, = +85.712; /a' = +79.712; / = + 15.497; /' = +20.712. 

7. / 2 = -66.67 cm. 

8. / = +20; at principal focal point F'\ 10 cm. outside of bowl. 

9. If the index of glycerin is taken as 1.48, / = 1.04 mm. 

10. If only one side of the cylinder is used, and considering r\ = +500 cm. 
and r 2 = +498 cm., / = -373,500 cm. Considering r { = r 2 = +500 
( . m .,/ = +750,000 cm. 

Chapter IV, page 34 

1. Less than 1 cm. 

2. Exit pupil is 4.8 cm. toward eye from ocular and has a diameter of 1 cm. 

3. HINT: find /3 for the entrance and exit pupils and use eq. 2-7. Image is 
20.9 cm. to right of exit pupil and is 1.31 mm. in height. 

4. If stop in front, //15; if behind, //1 3.5. 
6. 



Chapter V, page 44 

1. One point 135 cm. from 20-candle-power lamp toward 30-candle-power 
lamp; another point 133.5 cm. on side of 20-candle-power lamp. 

2. 1186. 

3. Brightness = 5.83 X 10~ 4 candles per cm. 2 = 1.83 X 10~ 3 lamberts. 

457 



458 LIGHT: PRINCIPLES AND EXPERIMENTS 

Chapter VI, page 70 

3. Distance of primary image from mirror vertex = 25.1 cm.; of secondary 
image = 36.2 cm. Length of primary image = 3.06 cm.; length of 
secondary image = 4.4 cm.; astigmatic difference = 11.1 cm. 

6. 8.31 and 12.85 along axis from vertex. 

6. fc = 50.0 179 cm., f = 50.000 cm., f P = 49.9645cm.,/,, = 50.0178cm. 

Chapter VII, page 86 

1. - 22.2. 4. ft = 8; / = 2.5. 

2. 0.67 mm. 5. 1 in. 

3. 4.16 cm. from glass. 7. /> = 7.79 cm.; 0* = 3.08. 

Chapter VIII, page 99 

2. Assuming minimum deviation in each case, dispersion at 4000 angstroms 
is 3.1 1 X 10 s radians per cm., at 7000 angstroms is 5.66 X 10 2 radians per 
cm. Sodium doublet just resolved if t 1.7 cm. 

4. About 65 deg. 5. n = 1.18. 

Chapter X, page 135 

1. 5.86 X 10~ 5 cm. 4. 5461 angstroms. 

2. n = 1.00029. 5. 5.76 X i()" 4 cm. 

3. About 14 17'. 

Chapter XI, page 162 

2. About 50 cm. 3. 0.0145 cm. 4. 5.945 mm. 

6. For R = 0.75, resolving power = 520,000; for R = 0.90, resolving 
power = 1,475,000. 

Chapter XII, page 206 

1. 1 .22 X 10~ 3 cm. 

2. For visual observations, about 51 sec. of arc or 3.5 mm. from center; for 
photographic observations, a larger distance. 

3. About 10,000 ft. 4. About 5.8 X 10~ 6 cm. 6. 6 = a. 

6. R = 150,000 in first order; about 113 cm. on a side; 3.567 mrn. 

Chapter XIII, page 249 

1. t = 0.015 times an odd integer; parallel to the optic axis. 

2. n = 1.58. 3. About 1 deg. 4. 0.0167T. 

5. Slightly elliptical, direction of major axis parallel to original plane of 
vibration. 

6. 1.99. 

Chapter XIV, page 271 

1. 9875.02, 15239.22, 18473.69, 20572.95, 22012.21, all cm- 1 . 

2. v x = 27419.42 cm.' 1 - )w - 3647 angstroms; V = 3.4 volts. 



ANSWERS TO PROBLEMS 459 

3. Total radiation per year = 1.42 X 10 41 ergs; of which 6.56 X 10" 1 strikes 
the earth. 

4. a, = 0.53 X I0-"cm.;t> = 2.182 X 10 8 cm. per sec.; W = 2.15 X 10~ 4 
ergs. 

6. Separations may be calculated from Rn/lti> = 8.9952/8.9976. 
6. T = 5786 deg. ahs.; E = 2.285 X 10 14 ergs. 

Chapter XVI, page 322 

1. About 15,000 gauss. 

2. Yes, at 2150 angstroms in first order, 4300 angstroms in second order, etc. 
4. About 20,000 gauss; R = 0.03. 

Chapter XVII, page 339 

1. Numerical aperture = 0.1503, assuming/' for eye is 2.07 cm 

2. About 30 cm. 3. Mngnification is about 3.33. 



INDEX 



Abbe, 59, 76 
Abbe condenser, 78 
Abbe number, 65 
Abbe refractometer, 99 
Abbe sine condition, 55 
Aberration, angle of, 114 

least circle of, 47 

of light, 113 
Aberrations, 45-70 

experiments in, 349-351 
Abney colorimeter, 334 
Abraham, 320 
Absorption, 272 

coefficients of, 273, 274 

continuous, 267 

laws of, 272 

line, 267 

arid radiation, law of, 251 

relation to dispersion, 276 

selective, 272-274 

and selective reflection, 281 
Absorption bands, 275, 276 

and index of refraction, 276, 279 
Absorption experiments, 361, 362 
Absorption index, 278, 279, 284 
Absorption maxima, table of wave- 
lengths of, 281 

Absorption spectra, 250, 266-268 
Absorptivity, 255 
Accommodation, 324 
Achromat, 67 

Achromatic combination, 65, 66 
Achromatism and focal length, 68, 

69 
Achromatizing of a thin-lens system, 

65, 66 

After-image, 333 
Air, index of refraction of, 444 
Allison, S. K., 299 



Aluminum for mirrors, 79, 433 

Ametropic, 325 

Amplitude, and intensity, 112, 113 

of single wave, 112 

of superposed waves, 110 
Analyzer, 221, 227, 396 
Anastigmat, 53 

Schwarzschild, 80, 81 
Anderson, J. A., 311 
Anderson, W. C., 322 
Angle, of aberration, 114 

critical, 96, 97 

of diffraction, 180, 181, 198, 200 

phase, 104, 105 

polarizing, 216-218 

of principal azimuth, 410 

of principal incidence, 410 
Angstrom, 64, 338, 443 
Angular magnification, 15 

of a telescopic system, 29 
Angular momentum, of electron, 
260, 261, 306 

of molecule, 264 
Anisotropic molecules, 289 
Anomalous dispersion, 275 
Anomalous Zeeman effect, 304, 305 

in zinc, illustrated, 305 
Anti-Stokes lines, 293, 294, 296, 297 
Aperture, 19, 44 

numerical, 41, 42, 43, 76 
of condenser, 77 
of microscope, 76 

relative, 35, 41 

of telescope, 80, 81 
Aperture stop, 31, 32, 34 

in correction, of astigmatism, 59 
of curvature of field, 59 

front, 32 

of a telescope, 79 
Aplanat, triple, 73, 74, 82 
Aplanatic lens system, 55 



461 



462 



LIGHT: PRINCIPLES AND EXPERIMENTS 



Aplanatic points, 55-58 

of microscope objective, 58 
Aplanatic surface, 4 

equation of, 4, 58 
Aplanatic system, 40 
Apochromatic system, 70 
Arago, 169 
Arkadiew, W ., 168 
Astigmatic difference, 52, 53 

measurement of, 350 
Astigmatic focal planes for meniscus 

lens, 60 

Astigmatic focus, primary, 393 
Astigmatic image positions, 51, 52 

equations for, 52 

derivation of, 424-426 

illustrated, 53 

Astigmatic spectral line, 393 
Astigmatism, 50-53 

correction of, 59 

of the eye, 325 

in paraholoidal reflectors, 80 
Atom model, Bohr's, 260 

Rutherford's, 259, 260 
Atomic number, 252 
Atomic spectra, 251-255, 260-264 

illustrated, 253, 267 
Avogadro number, 259, 269, 287, 289 
Axes, of elliptically polarized light, 

225 

positions of, 232, 400, 403 
ratio of, 403 

optic, in biaxial crystals, 240-242 

of single ray velocity, 241, 242 
Axis, optic, in uniaxial crystals, 209 
210, 242 

optical, 6, 8, 11 



B 



Babcock, //. />., 304, 306 

Babinet compensator, 226-232, 284, 

318 

Babinet-Jamin compensator, 227 
in analysis of elliptically polarized 

light, 401-403, 411 
Background, effects of, 43 
Balmer formula, 251, 252 



Buhner lines in Stark effect, 309 
Balmer series, in helium, 252 

in hydrogen, 251, 253, 254 
convergence X of, 271 
wave-lengths of, 443 
Balij, E. C. C., 395 
Band spectra, 264-268 

diagrams of, 266 

photographs of, 267 
Bartholinus, 208 
Beams, J. H'., 321 
Becquerel, Jean, 313, 314 
Beer's law, 273 
Benoit, J. R., 151 
Biaxial crystals, 215 

double refraction in, 237 

optic axes in, 240-242 

principal indices of refraction of, 
239 

rays in, 240 

wave surface in, 239, 240 
Bills, A. G., 331 
Binocular vision, 325 
Biprism (see Fresnel biprism) 
Birefringence, 319 
Birefringence constants, 319, 320 
Birge, It. T., 236 
Black body, defined, 255 
Blacks, absorbing, 275 
Blind spot, 323, 324 
Bohr, N., 101, 254, 255 
Bohr formula, 261, 262 
Bohr theory, of absorption and 
emission of radiation, 266-268 

of atomic spectra, 260-264 

of molecular spectra, 264-268 

in Raman effect, 295, 296 

of Stark effect, 311 
Boltzmann, 256 
Boltzmann's constant, 259 
Born, A/., 237, 282 
Brace, D. B., 313 
Brace-Lemon spectrophotomctcr, 

363-365 

Brace prism, 363, 364 
Brackett series, 254 
Bradley, 113, 114 
Bray, de, Gheury, 116 



INDEX 



463 



Brcit, (?., 277 

Brewster's law, 218, 233, 235 

verification of, 407-410 
Bright line spectra, 250-254 
Brightness, and color, 336, 337 

and the eye, 44, 327, 329 

of an image, 40, 42 
with the telescope, 43 

measurement of, 38 

natural, 41, 42 

of a reflecting surface, 38 

of the sky, 43 

of a star, 43, 44 
Brilliance, 331, 336 
Bunsen, 251 

C 

Cabannes, Jean, 287, 288 
Cadmium red line, 152, 155 
Calcite, 215 

crystal form of, 208, 209 

double refraction in, 208-211 

experiments with, 395-398 

indices of refraction of, 445 

optic axis of, 209, 210 
Campbell, N. /?., 315 
Canal rays, 310 
Candle, standard, 36 
Candle power, 36 
Cane sugar, optical rotation of, 244, 

405-407 
Cardinal points of a lens system, 16 

measurement of positions of, 347, 

348 

Cartesian oval, equation for, 58 
Cathodic sputtering, 434, 435 
Cauchy's dispersion formula, 63, 66, 
90, 275 

constants of, 99, 357 
Cellophane, 226, 398 
Chemical deposition of silver, 430 
Chief ray, 33, 34 
Chromatic aberration, 61-70 

axial, 65, 69 

illustrated, 62, 63 

longitudinal, 65, 351 

of plano-convex lens, measure- 
ment of, 351 



Chromatic difference of magnifica- 
tion, 65 

Chromaticity, 336 
Chromaticity diagrams, 337, 338 
Chromatism, lateral, 65 
Circles of confusion, 31 
Circular vibration, 303, 313 

angular velocity of, 314 
Circularly polarized light, 224, 225, 
228 

in Faraday effect, 314 

in metallic reflection, 410 

in rotatory polarization, 313 

in Zeeman effect, 303 
Coddington, H. y 52, 55 
Coddington eyepiece, 73 
Codding ton's shape and position 

factors, 49, 50 

Coefficient, of absorption, 273, 278 
molecular, 273 

extinction, 278, 279, 284 

of transmission, 274 
Coefficients, trichromatic, 336, 338 
Collimator, 85 

Col linear equations, for a single 
refracting surface, 11, 12 

for symmetrical coaxial systems, 

JO, 420 

Collinear relation, 10, 419, 420 
Collins, Mary, 327 
Collision broadening, 269 
Collisions of the first and second 

kind, 292 
Color, defined, 330 

dominant, 339 

and fluorescence, 289, 290 

and the retina, 331 

of the sky, 286 

surface, 273 

in thin films, 137 

transmitted, 274 
Color blindness, 331, 333 
Color diagram, three dimensional, 

337 

Color mixing, 334-338 
Color primaries, 334, 335 
Color sensitivity, 122, 331, 333 
Color triangle, 336, 337 



464 



LIGHT: PRINCIPLES AND EXPERIMENTS 



Color value, 336 
Color vision, 331 

theories of, 332-334 
Colorimeters, 334 
Colorimetry, standard sources for, 

436-438 
Colors, complementary, 332, 333 

primary, 334, 335 

in thin films, 137 
Coma, 53-55 

elimination of, 55 

illustrated, 54, 56 

observation of, 350, 351 

in paraboloidal reflectors, 80 
Comatic circles, 53, 54 
Combinations of two systems, 21 
Compton, A. //., 294, 299 
Condensers, 77 

Abbe, 78 

Cardioid, 78 
Conductivity, electrical, and optical 

properties, 282, 283, 284 
Cones, 331, 333 
Conical refraction, external and 

internal, 241, 242 
Conjugate points, 13, 14, 20 
Conjugate rays, 13, 14 
Continuous absorption and emis- 
sion, 267, 268 
Continuous spectra, 250 
Contrast sensitivity, 327 
Coordinates, in combination of two 
systems, 21 

in object and image spaces, 1 1 , 419 
Cornea, 323, 325 
Cornu double prism, 247 
Cornu-Jellet prism, 248 
Cornu polariscope, 222 
Cornu spiral, 171-176, 438, 440 

graph of, 441 

Cotton-Mouton birefringence con- 
stant, 319, 320 
Cotton-Mouton effect, 319 
Crew, H. 9 251 

Critical angle of refraction, 96, 97 
Cross-hairs, instructions for making, 

. 435, 436 
Crystalline lens, 323, 324 



Crystals, character of light trans- 
mitted by, 225 
classes of, 215 

direction of vibration in, 218, 219 
optic axes in, 209, 210, 240-242 
principal section of, 210, 219 
wave fronts in, 211 
wave surfaces in, 237, 239 
wave-velocity surfaces in, 211-215 

Curvature of field, 58 
correction of, 59 
experiment in, 351 

Cyanine, color transmission of, 274 



D 



Dark-field illumination, 77 

de Bray, Gheury, 116 

Descartes laws of refraction, 212 

Dichroism, 274 

Dielectric constant, 276, 277 

Diffraction, 164-206 

by circular opening, 203-206 
by rectangular opening, 202, 203 
(See also Fraunhofer diffrac- 
tion; Fresnel diffraction) 
Diffraction grating, 194-199, 390- 

395 

adjustments of, 390, 391 
concave, 393-395 
dispersion of, 197, 198, 393, 394 
mountings for, 393-395 
resolving power of, 198, 199, 392, 

393 

transmission, 393 
Diopter, 23 
Dioptric system, 27 
Dipole moment, 277, 289 
Direct-vision prism system, 96 
Direct-vision spectroscope, 95, 96 
Dispersion, anomalous, 275 
early theories of, 275, 276 
electromagnetic theory of, 277- 

279 

of a grating, 197, 198, 393, 394 
of a prism, in angstroms per 

millimeter, 92 
angular, 92 



INDEX 



465 



Dispersion, of a prism, factors affect- 
ing, 91 

at minimum deviation, 92, 358 
quantum theory of, 279, 280 
of rotation, 244 

Dispersion curve of a prism, deter- 
mination of, 357 

Dispersion formulas, 63, 275-278 
Dispersive power, 64, 65, 69, 356 
Distortion, 60 

experiment in, 351 
illustrated, 62 
Doppier broadening of a spectrum 

line, 269 
Double refraction, in calcite, 208- 

215 

in an electric field, 316-319 
in gases, 318, 319 
general treatment of, 237-240 
in liquids, 317, 318 
in a magnetic field, 319 
wave fronts in, 211-214 
wave surfaces in, 239 
Double slit interferometer, 387-390 
(See also Fraunhofer diffrac- 
tion ; Fresnel diffraction; 
Limit of resolution) 
Doublet, symmetrical, 61 
Drude, P., 55, 234, 236, 244, 277, 284 
du Bois, #., 316 
Du Bridge L., 236 



E 



e/m, 304, 306, 314 
Eagle mounting, 395 
Echelon, 199-202 

dispersion of, 200 

order of interference in, 200-202 

reflection, 202 

resolving power of, 201 
Einstein, A., 101, 260, 298 
Einstein's photoelectric equation, 

299 
Electric field, in Kerr effect, 317 

in Stark effect, 310 
Electric force, 233, 234 

and the light vibration, 234 



Electrical Kerr constant, 318 

for carbon bisulphide, 318 

table of, 319 
Electro-optical effect, 316 

in gases, 318, 319 

in liquids, 318, 319 

relation to electric field, 317 
Electron, angular momentum of, 
260, 261 306 

charge and mass of, 262 

ratio of charge to mass, 304, 306, 

314 

Electron spin, 269, 307 
Electronic bands, 266 
Elliptically polarized light, 223 

analysis of, 399-403 

analytic treatment of, 224, 229- 
232 

in Kerr effect, 315, 316, 318 

in metallic reflection, 284 

position of axes, 232, 400, 403 

ratio of axes, 403 
Emissive power, 255 
Emmetropic, 325 
Energy, internal, 264, 265 

rotational, 265 

Energy distribution, in interference 
patterns, 123, 124 

in standard sources, table, 437 
Energy distribution laws, 257-259, 

331, 362 
Energy-level diagrams, 262 

for absorption and emission, 268 

for anomalous Zeeman effect, 308 

for hydrogen, 263 

for normal Zeeman effect, 307 
Energy levels, in fluorescence, 292 

probability distribution in, 268 

for Raman effect, 295, 296 

splitting of, in magnetic field, 307, 

308 
Energy states, atomic, 262 

molecular, 264 
Entrance pupil, 32-34, 40 

of telescope, 79 

Evaporation method for coating 
mirrors, 431-434 



466 



LIGHT: PRINCIPLES AND EXPERIMENTS 



Exit pupil, 32, 33, 40, 42, 43 

of telescope, 79 

Extinction coefficient, 278, 279, 284 
Extraordinary ray, 395, 397 

in calcite, 211 

defined, 209 

direction of vibration of, in 'nicol, 
221 

indices of refraction of, 212, 445 
Extreme path, law of, 3, 4 
Eye, accommodation of, 324 

color sensitivity of, 122, 331, 333 

far and near points of, 324 

optical defects of, 324, 325 

optical system of, 323, 324 

schematic, 323 

sensitivity of, 327, 328 
Eye lens, 83 
Eyepiece, Coddington, 73 

erecting, 84, 85 

four-element, 84, 85 

Gauss, 74, 353, 354, 366, 427, 429 

Huygens, 69, 75, 82, 83 

micrometer, 74, 75 

Ramsden, 73, 74, 82-84 

triple aplanat, 73, 74, 83 
(See also Ocular) 

F 

f/ number, 35, 41 
Fabry, C., 151 

Fabry-Perot interferometer, 142, 
153-162 

fringe intensity distribution in, 
156-158 

fringe shape in, 161, 162 

order of interference in, 154, 155 

ratio of wave-lengths, determined 
with, 382-384 

resolving power of, 158-161 
Far point of the eye, 324 
Faraday, M., 300, 301, 312, 316 
Faraday effect, 312 

experiment in, 414-416 

explanation of, 313 

and Kerr effect, 316 

magnitude of, 314 

in solutions, 314 



Format's principle, 3-5 

Field lens, 83 

Fine structure of spectrum lines, 

151, 152, 202, 269, 312 
Fizeau, 114-116 
Flicker photometer, 328 
Flicker sensitivity, 327, 328 
Fluorescence, of atoms, 292 

in gases, 291, 292 

polarization of, 290 

and Raman effect, 297 
Focal distances, astigmatic; deriva- 
tion of equations for, 424-426 
Focal length, of combination of 
two thin lenses, 23 

equivalent, 7 

of thin lens system, 6, 20 

from power formula, 25, 26 
Focal length measurement, 343 

by autocollimation, 344 

of divergent lens, 345 
Focal lengths, of components of an 
achromat, 66 

principal, of a system, 14 
Focal points in combinations, 21 
Focus, meridional, 51 

primary, 51, 393, 426 

sagittal, 51 

secondary, 51 

tangential, 51, 426 
Foster, E. W., 435 
Foster, J. S., 310, 311 
Foucault, 114-116 
Fovea, 323, 324, 331 
Fowler, A., 253 
Frank, N. H., 277 
Fraunhofer, 251 
Fraunhofer diffraction, 176, 385 

defined, 164 

in the echelon, 199-202 

illustrated, 179, 196, 203, 206 

by single slit, 178, 180-184 

by two slits, 179, 184-186, 387- 

390 
Fraunhofer lines, 64, 356 

table of, 64, 443 
Fred, M., 267 
Fresnel, 111, 165, 169, 237 



INDEX 



467 



Fresnel biprism, 126, 129-132 

interference fringes with, 131 

in wave-length determination, 
368-370 

wave-length equation for, 130, 132 
Fresnel diffraction, 176 

denned, 164 

illustrated, 168, 171, 177 

by single slit, 175 

by straight edge, 169-171, 174 

by two slits, 176, 177 
Fresnel equations, 233-235, 282 

for absorbing media, 283 
Fresnel integrals, 438-441 

table of, 440 
Fresnel mirrors, 126-128 

interference fringes with, 128 

in wave-length determination, 370 

wave-length equation for, 127 
Fresnel rhomb, 236, 237 
Fresnel zones, 165, 166, 287 
Fresnel's theory of rotatory polariza- 
tion, 244-247 



G 



Gaertner Scientific Company, 95, 395 

Gate, H. G., 267, 270 

Gardner, I. C., 97 

Gauss eyepiece, 74, 353, 354, 366, 

427, 429 

Gaussian image point, 46, 47 
Geometrical optics, postulates of, 1 
Ghosts, Rowland, 253, 391 

Lyman, 391 
Gilbert, P., 440 

Glan polarizing prisms, 359, 363 
Glan spectrophotometcr, 358, 359 
Grating (see Diffraction grating) 
Grating mountings, 393-395 
Group velocity, 117-119 
Guild, J., 436 

H 

h, 259, 262, 299, 311 
Hagen, E., 284 
Hale, G. E., 301 



Half-shade plates and prisms, 248, 

404 

Half-wave plate, 223, 226, 397 
Half-width of a spectrum line, 269 
Hardy, A. <?., 70, 332, 336, 338 
Harrison, G. R., 267 
Hefner lamp, 36 
Heisenberg, W ., 280, 295 
Helmholtz, 276, 277, 327, 332 
Helrnholtz-Ketteler dispersion for- 
mula, 276 

Hering's theory, 333 
Houstoun, R. A., 112, 277, 302 
Hue, 330 

Huff or d, M. E., 168 
Huygens, 208, 219 
Huygeiis construction in double 

refraction, 211-213 
Huygens ocular, 69, 75, 82, 83 
Huygens principle, 110-112, 165, 166 
Huygens wavelets, 111, 210, 211 
Hydrogen, Stark effect in, 309 

visibility curve for H a of, 152 
Hydrogen series, 253, 254 

wave-lengths of, 443 
Hyperfine structure, 270 
Hyperopia, 325 

Hyposulphite of soda, light scatter- 
ing by, 285, 412, 413 



Iceland spar (sec Calcite) 
llluminant, standard, 336, 338, 436- 

438 
Illumination, of an image, 39-41 

of a surface, measurement of, 37 
Illuminators, dark field, 77 

vertical, 77 
Index of absorption, 278, 279, 284 

measurement of, 411 
Index of refraction, of carbon bisul- 
phide, 119 
complex, for absorbing media, 278, 

283 

defined, 2 

determined, by minimum devia- 
tion, 89 



468 



LIGHT: PRINCIPLES AND EXPERIMENTS 



Index of refraction, determined, by 

refractometer, 133-135 
by total reflection, 97, 98 
in double refraction, 212 
in eletromagnetic theory, 277, 

278 

measurement of, by Babinet com- 
pensator, 411 
for divergent lens, 345 
of glass block, 352 
of glass plate, 367, 368 
of glass prism, 365, 366 
of liquids, 352, 353, 367 
by Michelson interferometer, 

376-380 

by microscope, 352, 353 
by spherometer, 345 
by total reflection, 365-368 
for metals, 284 
Indices of refraction, in Faraday 

effect, 314 

numerical values of, 444, 445 
principal, 239 

Intensity comparisons, 37, 359 
of bright lines 362 
of continuous spectra, 360, 301 
Intensity distribution, in continuous 

spectra, 257-259, 360 
in diffraction patterns, 170, 171, 

176, 178, 183, 185, 195,205 
in Fabry-Perot interferometer 

fringes, 156-158 

in interference patterns, 123, 124 
Interference, compared with diffrac- 
tion, 120, 176 
conditions for, 120-123 
division of amplitude, 124, 125, 

137-162 

in Newton's rings, 137-141 
in thin films, 137 

(See also Fabry-Perot inter- 
ferometer; Michelson inter- 
ferometer) 

division of wave front, 124, 125 
diffraction in, 176 
Fresnel biprism, 126, 129-132 
in Fresnel mirrors, 126-128 



Interference, division of wave front, 
in Rayleigh refractometer, 
126, 132 

(See also Echelon) 
with double and multiple beams, 

142 
general methods for production 

of, 124, 125 

order of, 154, 155, 198, 199 
Interference fringes, in diffraction 

patterns, 176-179, 185 
forms of, 146, 147, 161, 162 
illustrated, 128, 131, 138, 141, 

149, 150, 155, 177, 179 
intensity distribution in, 156-158 
visibility of, 147-152, 158, 177, 

179, 187 

defined, 148, 157 

Interferometer, double slit, 387-390 
Fabry-Perot, 142, 153-162 
Michelson, 142-152, 370-382 
stellar, 153, 187-194 
loriization, 267 
Irradiation, 44, 326 
Isotope effect, 270, 312 
Isotropic medium, 2 
refraction in, 212 
Ives, H. E., 334 
Ives colorimeter, 334 



Jamin, 217 

Jeans, Sir J., 258 

Jena Glass Works, 59, 70, 362 

Jones, E. G., 435 



K 



Karolus, A., 321 
Katoptric system, 27 
Kayser, 77., 443 
Kepler, 250 
Kerr, 315, 316 
Kerr cell, 318 

in measurement, of time intervals, 

320, 321 
of velocity of light, 321, 322 



INDEX 



469 



Kerr constant, electrical, 318 

magnetic, 316 
Kerr effect, electro-optical, 316-319 

magneto-optical, 315 
KeMeler, 276 
Kirchhoff, 251, 255 
KirchhofPs laws of absorption, emis- 
sion, and radiation, 255 
Korff, S. A., 277 
Kramers, IL A., 280, 294, 295 
Krishnan, K. N., 293 
Kundt, 314 



L 



Ladd-Franklin theory, 333 
LaGrange's law, 14, 55 
Lambert, the, defined, 38 
Lambert's cosine law, 38, 39 
Lambert's law of absorption, 273 
Langsdorf, A., Jr., 236 
Larmor, /., 315 
Lateral magnification, 43, 54 

defined, 11 

of a telescopic system, 28 
Lateral spherical aberration, 48, 50 
Laurent half-shade plate, 248, 404 
Laurent polar i meter, 404 
Lawrence, E. O., 321 
Least time, principle of, 3 
Lemoine, J., 320 
Lens (see Thin lens) 
Lens combination, zero power, 81 
Lens combinations, 21-29, 65-67 
Lens formula, general; derivation 

of, 23-26 

Light, theories of, 100-102 
Light flux, unit of, 36 
Limit of resolution, of one and two 
slits, 186, 187 

Rayleigh's criterion of, 186 

(See also Resolving power) 
Line spectra, absorption, 260 

emission, 250-254 
Littrow mounting, 247 
Lockyer, Sir N., 251 
Longitudinal chromatic aberration, 
63, 351 



Longitudinal magnification, 15 
of a telescopic system, 28 

longitudinal spherical aberration, 48 
measurement of, 349, 350 

Lorentz, H. A., 277, 301, 302 

Lo Surdo, 309-31 1 

Lumen, defined, 36 

Luminous intensities, comparison of, 
37 

Luminous intensity, of an image, 41 
of a source, 36 

Lummer, O., 258 

Lyinan, T., 253 

Lyman ghosts, 391 

Lyrnan series, 253 



M 



Me Donald observatory telescope, 80 
Magnetic field, angle of rotation of 
plane of vibration produced by, 
313, 416 

effect of, on energy levels, 307, 308 
on light source, 300 

strength of, in Faraday effect, 416 
in Zeeman effect, 302, 304, 308 
Magnetic force, 234 
Magnetic Kerr constant, 316 
Magneto-optical effect, 315, 316 
Magnification, angular, 15, 22, 29 

chromatic difference of, 65 

of compound microscope, 75 

lateral, 11, 28, 54 

longitudinal, 15, 28 

normal, 43 

of simple microscope, 72 
Magnifier, compound, 72, 73 

simple, 72 

(See also Eyepiece; Ocular) 
Magnifying power, 29, 78 
Mains, 215 

cosine-square law of, 220 

law of, 5 

Mann, C. R., 146 
Martin, A. E., 141 
Mass, reduced, 262 
Maxwell, J. f .,102 



470 



LIGHT: PRINCIPLES AND EXPERIMENTS 



Metallic reflection, experimental 

facts of, 410 

Fresnel equations for, 283 
of polarized light, 315, 316 
Metals, contrasted with transparent 

substances, 282 
optical constants of, 284, 410, 411 

table of, 412 , 
reflecting powers of, table, 445 
Meyer, C. F., 176, 205 
Mica, 226 
Michelson, A. A., 116, 117, 119, 144, 

151, 152, 199, 322 
Michelson interferometer, 142, 143- 

152 

adjustment of, 370-372 
form of fringes in, 146, 147 
measurement, of distance with, 

370-376 
of index of refraction with, 376- 

380 

resolving power of, 143, 151, 153 
visibility of fringes in, 147-152 
wave-length ratios with, 380-382 
Michelson stellar interferometer, 

153, 187-194 

Micrometer eyepiece, 74, 75 
Microscope, compound, 75 
numerical aperture of, 76 
oil-immersion, 76 
simple, 72, 73 
Microscope objective, aplanatic 

points of, 58 
Millimicron, 338 
Mills, John, 313 

Minimum deviation of a prism, 89 
and dispersion, 92 
and resolving power, 93 
Mirror, paraboloidal, 79, 80 

aberrations of, 80 
Mirror surfaces, preparation of, 430- 

435 

Mirror systems, equations for, 17 
Mittelstaedt, 0., 321, 322 
Molecular absorption coefficient, 273 
Molecular rotation of plane of vibra- 
tion, 244 
Molecular spectra, 264-268 



Molecules, aniso tropic, 289 

Monk, G. S., 267 

Monochromator, 95 

Atulliken, R. S., 267 

Multiplet structure, 269, 270, 305, 

312 
Myopia, 324 

N 

n slits, diffraction by, 194, 196 
Near point of the eye, 324 
Negative crystals, 215 

uniaxial, 242 
Negative lens system, 27 
Newton, Sir I., 251 
Newtonian telescope*, 81 
Newton's rings, 137-141 

illustrated, 141 

wave-length equation for, 140 
Nichols, E. F., 280, 281 
Nicol, W., 220 
Nicol prism, construction of, 220, 

221 

Night blindness, 331 
Nodal points, defined, 16 
Nodal .slide, 347 
Normal magnification, 43 
Normal triplet, 340 
Nuclear spin, 270, 312 
Numerical aperture, 41-43 

of condensers, 77 

of microscope, 76 

O 

Objective, 28, 29 
Ocular, 28, 29 
negative, 82 

(See also Eyepiece) 
Optic axes in biaxial crystals, 240- 

242 

Optic axis, 242 
of calcite, 209 
defined, 210 
Optical axis, 6, 8, 1 1 
Optical constants of metals, 284, 

410, 411 
table of, 412 



INDEX 



471 



Optical illusions, 326, 327 

Optical rotation, in cane sugar, 

244, 405-407 
in quartz, 243, 405 
Optical system, ideal, 8, 419 

equations for, 420 
symmetrical, properties of, 419, 

420 
Optically active substances, 243, 

312, 313 
molecular, or specific, rotation by, 

244 
Order of interference, 154, 155, 198- 

202 

Ordinary ray, in culcite, 211 
defined, 209 

index of refraction for, 212, 445 
Orthoscopic system, 61 
Orthotomic system, 5 



Parallel (IT) components in Zeeman 

effect, 305, 306 
Paraxial ray, 14 
Parsons, J. H., 327, 332 
Paschen-Back effect, 308, 309 
Paschen mounting, 394 
Paschen series, 254 
Pearson, F. t 117 
Pease, F., 117 
Perot, A., 151 
Perpendicular (a-) components in 

Zeeman effect, 305, 306 
Perrin, F. H., 70 
Petzval condition, 59. 70 
Pfund series, 254 
Phase angle, 104, 105 
Phase change on reflection, 139, 234 
Phase difference, 104, 107, 109, 124, 

145, 156, 180, 195, 402, 403 
Phase retardation, 139 

in Babinet compensator, 228 

in crystals, 223, 224 
Phosphorescence, 291 
Photoelectric effect, 298, 299 
Photometer, 327 

flicker, 328 

(See also Spectrophotometer) 



Photometric standards, 36 
Photometry, 37 
Photon, 100, 299 
Pigment mixing, 334 
Pin-hole optical system, 60 
Planck, M., 101, 258 
Planck's constant, 437 
Planck's distribution law, 259 
Planck's quantum constant, h, 259, 

262, 299, 311 
Planck's quantum hypothesis, 258, 

259, 298 

Plane, of polarization, 219, 284 
of vibration, 219, 284, 397 

rotation of, by electric field ,- 

316-319 
by magnetic field, 312-blo, 

319, 320 

by optically active sub- 
stances, 242-247, 405-407 
on reflection, 234, 235, 407-40CJ 
Plane polarized light, 217, 219, 220, 
222, 223 ' 

in metallic reflection, 410 
passage of, through o crystal, 224, 

225 

plane, of polarization of, 219, 284 
of vibration of, 219, 284, 397 
(See also Plane of vibration) 
in Zeeman effect, 304 
Poisson, 169 
Polarimeter, 403 

Laurent, 404 
Polariscope, 395, 396 

Conui, 222 
Polarization, 208-249 

circular, 224, 225, 228, 303, 313, 

314, 410 
by double refraction, 208-215, 

237-240 

elliptical, 223-232, 284, 315 
of fluorescence, 290 
plane of, 219, 284 
by reflection, 215-218 
by refraction, 216, 217 
rotatory, 242-249, 312, 313, 403- 

407 
of scattered light, 287-289, 412-414 



472 



LIGHT: PRINCIPLES AND EXPERIMENTS 



Polarized light, reflection of, 232- 
235, 410 

in spectrophotometers, 358jf 
Polarizer, 221, 227, 396 
Polarizing angle, 216, 217 

of glass, 216, 218 
Polarizing prisms, Cornu-Jellet, 248 

Glan, 359, 363 

nicol, 220 

Wollaston, 221 
Polaroid, 215, 395 
Pole of wave front, 169 
Position factors, 49, 50 
Positive crystals, 215 

uniaxial, 242 
Positive lens system, 27 
Power, candle, 36 

dispersive, 64, 65, 69, 356 

emissive, 255 

of a lens, or lens system, 6, 23, 24, 
26 

magnifying, 29, 78 

reflecting, 160, 445 

resolving, 93, 151, 158-160, 186, 

198, 199, 206, 357, 358 
x> rf w-,..- effect in spectrum lines, 269 
Preston, T., 112, 115, 176, 205, 237, 

276 
Primary standard of wave-length, 

152, 155 

Principal focal lengths, 14 
Principal focal points, of coaxial 
optical systems, 1 1 

of a spherical surface, 9 

of a thin lens, 6 
Principal planes, 6, 7 

denned, 12 

of a telescopic system, 28 
Principal points, denned, 12 
Principal section, of a crystal, 210, 
219, 397 

of a nicol, 221 
Pringsheim, P., 258 
Prism, dispersion of, 89-92, 358 

dispersion curve of, 356 

measurement of index of refrac- 
tion of, 355, 356, 365, 366 

minimum deviation of, 89, 355, 356 



Prism, refracting angle of, 88 
measurement of, 354, 355 

resolving power of, 92-94, 357, 358 

total deviation of, 88 
Prism binocular, 84 
Prisms, Brace, 363, 364 

constant-deviation, 94 

total-reflection, 96, 97, 193, 358 

(See also Polarizing prisms) 
Purity, denned, 274 
Purkinje effect, 328, 329 



Q 



Quantum, of energy, 100 

of radiation; relation to wave- 
front, 299 
Quantum constant, h, 259, 262, 299, 

311 

Quantum mechanics, 101 
Quantum numbers, 265, 307 
Quantum theory, 101 

Planck's, 258, 259 

of spectra (see Bohr theory) 
Quarter-wave plate, 224, 226, 227, 

284,397-401,411 
Quartz crystals, 215, 247, 248 

indices of refraction of, 445 

optical rotation of, 243, 405 



R 



Radiation, and absorption, law of, 

251 

resonance, 292, 293 
Radiation laws, 255-259, 331, 362 
Radius of curvature, of concave 

surface, measurement of, 346 
of cornea, 323 
measurement of, with npherom- 

eter, 345 
sign of, 8, 19 

Raman, C. F., 287, 293, 294 
Raman effect, 293-298 
in carbon tetrachloride, illus- 
trated, 293 
Raman lines, 295 
intensities of, 297 



INDEX 



473 



Ramsden eyepiece, 73, 74, 82-84 
Ray, in biaxial crystals, 240-242 
defined, 2 

optical length of, 2, 3 
Rayleigh, Lord, 94, 117, 160, 286 
Rayleigh , Lord (the younger; R. J. 

Strutt), 289, 291 
Rayleigh distribution law, 258 
Rayleigh-Jeans law, 258, 259 
Rayleigh refractometer, 132-135 
Rayleigh 's criterion, of limit of 

resolution, 186-188 
of resolving power, 93, 94, 198, 357 
Rectilinear system, 61 
Reduced mass, 262 
Reflecting power, of metals, table of, 

445 

and resolving power, 160 
Reflecting surface, aplanatic, 58 

brightness of, 38 
Reflecting telescope, 80-82 
Reflection, diffuse, 38, 77 

Fresnel equations for, 233-235, 

282, 283 

metallic, 282-284, 315, 316, 410 
plane of, 220 
of polarized light, 232-235, 283, 

284 
rotation of plane of vibration by, 

234, 235, 407-409 
selective, 280, 281 
specular, 77 
Reflectivity, 281 
defined, 282 
equations for, 283, 284 
measurement of, 411 
Refracting surface, aplanatic, 58 
astigmatic focal distances for, 424, 

426 

colliriear equations for, 1 1 
Refracting telescope, 78, 79, 84 
Refraction, external and internal 

conical; 241, 242 
laws of, 1, 212 

for extraordinary ray, 212-214 
at a spherical surface, 8-10 
(See also Double refraction) 



Refractometer, 365 

Abbe, 99 

Rayieigh, 132-135 

Williams, 135 
Reiche, F., 255 
Relative aperture, 35, 41 

of a telescope, 80, 81 
Residual rays, 280, 281 

table of, 281 

Resolving power, of a circular 
opening, 206 

of Fabry-Perot interferometer, 
158-160 

of a grating, 198, 199 

and limit of resolution, 186 

of Michelson interferometer, 143, 
151, 153 

of one and two slits, 186, 187 

of a prism, 92-94 

measurement of, 357, 358 
Resonance radiation, 292, 293 
Retina, 324, 331 
Reversibility, principle of, 5 
Richtmyer, F. K., 255, 257, 299 
Righi, A., 313 
Rite, H 7 ., 253 
Rods, 331, 333 
Romer, 113 
Rosette orbit, 303 
Ross, F. #., 81 
Rotating mirror, in velocity of light 

measurement, 114-117 
Rotation, dispersion of, 244 

of plane of vibration (see Plane 
of vibration) 

molecular, 244 

specific, 244 
Rotatory polarization, 242-249 

of common substances, 403-407 

contrasted with Faraday effect, 
312 

explanatipn of, 313 

FresnePs theory of, 244-247 
Rowland circle, 393, 394 
Rowland ghosts, 253, 391 
Rowland mounting, 394 
Ruark, A. E., 311 
Rubens, 280, 284 



474 



LIGHT: PRINCIPLES. AND EXPERIMENTS 



Runge, C. t 305 

Rutherford's atom model, 259, 260 

Rydberg, J. R., 253 

Rydherg number, 252, 262 



Sabine, G. B., 435 

Saccharimeter, 404 

Saturation, 330, 331 

Scattered light, intensity of, 287, 288 

polarization of, 287-289, 412-414 
Scattering, coherent, 298 

by gases, 285-289 

incoherent, 298 

by liquids, 293 

Raman, 293-298 

secondary, 289 

of x-rays, 294 
Scheiner, J., 251 
Schmidt, B., 81, 82 
Schmidt corrector for telescope, 81, 

82 

Schuster, A., 166, 237, 286 
Schuster's method of focussing, 428 
Schwarzschild anastigmat, 80, 81 
Secondary waves, 110, 111 
Selection principles, 265, 307 - 
Sellmeier's dispersion formula, 275, 

276 

Sensitivity of the eye, for color, 122, 
131, 333 

for contrast, 327 

flicker, 327, 328 

spectral, 328, 329, 333 
Series in spectra, 251-254, 267 
Shade, 329 
Shape factors, 49, 50 
Shedd, J. ., 146 
Sign conventions, for A, 21 

for ideal optical systems, 8, 11,21, 
419 

for mirrors, 17 

for radius of curvature, 8, 19 

for single refracting surface, 8 
Silver, cathodic sputtering of, 434, 
435 

chemical deposition of, 430 



j tolver, reflecting power of, table, 445 
i Simple harmonic motions, 103-105 

composition of, analytical, 107, 

108 

graphical, 105, 106 
Single slit (see Fraunhofer diffrac- 
tion; Fresnel diffraction; 
Resolving power) 
Sky, color of, 286 
Slater, J. ., 277 
Sleator, W. W., 141 
Smekal, A., 294, 295 
Smith, T., 436, 443 
Smith-Helmholtz law (see La- 
grange's law) 
' Snell's law, 1 

at the polarizing angle, 218 

for small angles, 14 
Sodium absorption, 267 
Sodium doublet, Zeeman effect in, 

308 

Sodium series, 253, 267 
SouthaU, J. P. C., 327 
Space quantization, 307 
Specific rotation, 244, 247, 406 
Spectra, 250-271 

band, 264-268 

Bohr theory of, 260-262 

classification of, 250 

multiplets in, 269 

quantum theory of, 254, 255, 260- 
271 

resonance, 293 

series in, 252-254 
Spectral sensitivity, 328 
Spectral transmission, 362 
Spectrograph, 84 
Spectrometer, 84-86 

adjustment of, 426-430 

constant-deviation, 94, 95 

parts of, 85 

prism, 88/, 353-358 
Spectrophotometer, 358-365 
Spectrophotometry, 37 
Spectroscope, direct-vision, 95, 96 
Spectrum, 63, 329 

secondary, 67, 69 

solar, 64, 443 



INDEX 



475 



Spectral lines, astigmatic, 393 

breadth of, 268, 269 

curved, 92 

fine structure of, 151, 152, 202, 
269, 312 

half-width of, 269 

hyperfiiie structure of, 270 

pressure effect in, 2G9 

Stark effect on, 271, 309-312 

Zeeman effect on, 270, 300-309 
Spherical aberration, 46-50 

condition for elimination of, 55 

correction of, 47 

illustrated, 46, 47 

lateral, 48, 50 

of lens combinations, 50 

longitudinal, 48, 349, 423 
measurement of, 349 

of plano-convex lens, 47, 50, 343, 
350 

third order corrections for, 48 

derivation of, 421-423 
Spherometer, 344 
Standard candle, 30 
Standard illuininant, 336 

chrornaticity of, 338 
Standard sources for colorimetry, 
436-438 

energy distribution in, table, 437 

(See also Illuminant) 
Standard wave-lengths, 152, 155, 156 
Standing waves, 234 
Stark effect, 309-312 

apparatus for, 309, 310 

Bohr theory of, 311 

illustrated, 310 

transverse, 310 
Stefan-Boitzmann law, 256 
htellar diameters, measurement of, 

189, 192, 193 
Stereoscope, 325, 326 
Stokes' law, 290 
Stokes lines, 293, 294, 296, 297 
Stop, effect of, 31 

(See also Aperture stop) 
Strong, /., 435, 445 
Strutt, R. J. (Lord Rayleigh, the 
younger), 289, 291 



Sun, magnetic field in, 301, 302 
Sunspots, Zeeman effect in, 301 
Superposition, of fringe systems, 
149, 150 

principle of, 109 

of two waves, 109, 110, HI, 123 
Surface color, 273 
Symmetrical optical system, 419, 420 



Table, of birefringence constants, 

319 

of complementary colors, 332 
of current and temperature cali- 
bration of standard lamp, 363 
of Fraunhofer lines, 64, 443 
of Fresnel integrals, 440 
of indices of refraction, 444, 445 
of logarithms, 446, 447 

of trigonometric functions, 452- 

455 

of natural trigonometric func- 
tions, 448-451 

of optical constants of metals, 412 
of TT-, a- components of a zinc 

multiplet, 306 
of positions, of diffraction minima 

for circular opening, 205 
of single slit maxima, 183 
of reflating power of metals, 445 
of spectral distribution of energy 

in standard illuminants, 437 
of Verdet's constant, 314 
of wave-lengths, of absorption 

maxima, 281 
of various elements, 443 
Talbot's law, 328 
Taylor, H. D., 48 
Teleceutric systems, 34 
Telescope, 28 

entrance and exit pupils of, 79 
magnifying power of, 78 
reflecting, 80-82 
refracting, 78, 79, 84 
of spectrometer, 85 
Telescopic system, 27-29 
Theories of light, 100-102 



476 



JGHT: PRINCIPLES AND EXPERIMENTS 



TV' d, colors in, 137 

<3ns, derivation of equation 
for, 19, 20 

focal length of, 6, 20 
optical axis of, 6 

positions of astigmatic foci for, 52 
spherical aberration of, 48, 50, 

421-423 

Thin lens system, 19-29 
"acfirbmatic combination, 65, 66 
focal length of, 6, 23, 26 
Tint, 329 
Toothed wheel, in velocity of light 

measurement, 114 
Total reflection, 96 
Total-reflection prism, 96, 97, 193 
Tourmaline, 215 

index of refraction of, 445 
Transverse vibration, direction of, 

in crystals, 218, 219 
and the electric force, 234 
evidence for, 217 

Trichromatic coefficients, 336, 338 
Trichromatic theory of Young- 

Helmholtz, 332, 333, 334 
Tristimulus values, 336 
Troland, L. T., 327 
Twyinan, F. TV., 443 
Tyndatt, J., 286 



U 



Uniaxial crystals, 215, 219, 226, 242, 

243 
experiments with, 395, 397, 398, 

405 

indices of refraction of, 445 
Unit planes (see Principal planes) 
Unpolarized light, nature of, 235 
Urey, H. C., 311 



Valasek, J., 285 
van Biesbroeckj G. -A., 56 
Velocity of light, in carbon bisul- 
phide, 119 
in crystals, 210, 222 



Velocity of light, determinations of, 
] 113-117 

' with Kerr cells, 321, 322 
relation of, to frequency and 

wave-length, 102 
wave and group, 117-119 
Verdet's constant, for carbon disul- 

phide, 313 
defined, 313 
measurement of, 415 
table of, 314 
Vertical illuminators, 77 
Vibration, plane of (see Plane of 

vibration) 
Visibility of interference fringes, 177, 

179, 385 

analysis of, 189-191 
defined, 148, 157 
in Fabry-Perot interferometer, 

158 ' 
in Michelson interferometer, 147- 

152 

minimum, 187, 389 
in test for limit of resolution, 187 
Visibility curves, 151, 152 
Vision, binocular, 325 
color, 331-334 

functions of rods and cones in, 331 
persistence of, 327, 328 
stereoscopic, 326 
Vaigt, W., 277 
von Seidel, L M 46 



W 



Wave front, 110 

in crystals, 211-214, 237-240 
cylindrical, 169 
pole of, 169 

Wave and group velocity, 1 17-1 19 
Wave-length determination, by 

Fresnel biprism, 368-370 
by single slit diffraction, 384-387 
Wave-length standards, primary, 

152, 155 
secondary, 156 

Wave-lengths, of absorption max- 
ima, table of, 281 



INDEX 



477 



Wave-lengths, ratio of, with Fabry- 
Perot interferometer, 382-384 
with Michelson interferometer, 
380-382 

of various elements, table of, 443 
Wave motion and light, 100, 102 

(Characteristics of, 108, 109 

displacement in, 108, 109 

velocity of, 108 
Wave-number, defined, 252 
Wave plate, 223, 227 
Wave surface in biaxial crystals, 239 
Wave-velocity surface, 211-215 

of calcite, illustrated, 214 
Weierstrass, 57 
White, H. L\, 253, 311 
Whittaker, E. T., 46 
Wieii displacement laws, 256 
Wien distribution law, 257, 331, 362 - 
Wiener, 234 
Williams, R. C., 435 
Williams, W. E., 135, 143, 156 
Williams refractorneter, 135 
Woliaston prism, 318, 358, 360 

construction of, 221, 222 
Wood, R. W., 168, 275, 292, 297, 362 
Woodworth, C. W., 23 



Yourig, Thomas, 57, 332 
Young-Helmholtz theory, 332, 333, 

334 

Young's apparatus, 126, 132 
Young's construction, 57 
Young's experiment, 125, 126 



Z 



Zeeman effect, 300-309 

anomalous, 304, 305, 306-308 
in chromium, illustrated, 304 
classical theory of, 302-304 
energy levels in, 307, 308 
inverse, 301 
, normal, 305, 307 
quantum theory of, 306-309 
in sunspots, illustrated, 301 

Zeeman patterns, 304 

anomalous, in sodium, 308 

in zinc, 305 
normal triplet, 304 

Zero power lens combination, 81 

Zone plate, 166, 167 

Zones, Fresnel, 165, 166